Slide 40 (Answer) / 309 15 What is the x-intercept of this line? 10 8 6 Answer 4 (-4,0) 2 0 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 [This object is a pull tab] -6 -8 -10 Slide 41 / 309 16 What is the x-intercept of this line? 10 8 6 4 2 0 -8 -10 -6 -4 -2 2 4 6 10 8 -2 -4 -6 -8 -10 Slide 41 (Answer) / 309 16 What is the x-intercept of this line? 10 8 Answer 6 (3,0) 4 2 0 -10 -8 -6 -4 -2 2 4 6 8 10 -2 [This object is a pull tab] -4 -6 -8 -10
Slide 42 / 309 17 What is the x-intercept of this line? 10 8 6 4 2 0 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 Slide 42 (Answer) / 309 17 What is the x-intercept of this line? 10 8 6 Answer 4 (-6,0) 2 0 -8 -4 -2 -10 -6 2 4 6 8 10 -2 -4 [This object is a pull tab] -6 -8 -10 Slide 43 / 309 18 The graph of the equation x + 3y = 6 intersects the y-axis at the point whose coordinates are A (0,2) B (0,6) C (0,18) D (6,0) From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Slide 43 (Answer) / 309 18 The graph of the equation x + 3y = 6 intersects the y-axis at the point whose coordinates are A (0,2) B (0,6) C (0,18) Answer D (6,0) A) (0,2) [This object is a pull tab] From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. Slide 44 / 309 Defining Slope on the Coordinate Plane Return to Table of Contents Slide 45 / 309 "Steepness" and "Position" of a Line
Slide 46 / 309 Consider this... An infinite number of 10 lines can pass through the same location on the 8 y-axis...they all have the 6 same y-intercept. 4 Examples of lines with a 2 y-intercept of ____ are shown on this graph. 0 -10 -8 -6 -4 -2 2 4 6 8 10 What's the difference -2 between them (other -4 than their color)? -6 -8 -10 Slide 47 / 309 The Slope of a Line The lines all have a 10 different slope. 8 Slope is the steepness 6 of a line. 4 Compare the 2 steepness of the lines on the right. 0 -8 -2 -10 -6 -4 2 4 6 8 10 -2 Slope can also be -4 thought of as the rate of change. -6 -8 -10 Slide 48 / 309 The Slope of a Line run 10 8 rise The red line has a positive slope, since 6 the line rises from left 4 to the right. 2 0 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10
Slide 49 / 309 The Slope of a Line 10 The orange line has a 8 negative slope, since 6 the line falls down from left to the right. 4 rise 2 run 0 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 Slide 50 / 309 The Slope of a Line The purple line has a 10 slope of zero, since it doesn't rise at all as 8 you go from left to 6 right on the x-axis. 4 2 0 -8 -2 -10 -6 -4 2 4 6 8 10 -2 -4 -6 -8 -10 Slide 51 / 309 The Slope of a Line 10 The black line is a vertical line. It has an 8 undefined slope, since 6 it doesn't run at all as 4 you go from the bottom to the top on the y-axis. 2 0 -8 -10 -6 -4 -2 2 4 6 10 8 rise -2 = undefined 0 -4 -6 -8 -10
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Slide 58 / 309 Slide 59 / 309 Slide 60 / 309 Measuring the Slope of a Line run While we can quickly 10 see if the slope of a rise line is positive, 8 negative or zero...we 6 also need to determine 4 how much slope it has...we have to 2 measure the slope of a 0 line. -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10
Slide 61 / 309 Measuring the Slope of a Line The slope of the line is 10 run just the ratio of its rise 8 over its run. rise 6 The symbol for slope 4 is "m". 2 So the formula for 0 slope is: -10 -8 -6 -4 -2 2 4 6 8 10 -2 slope = rise -4 run -6 -8 -10 Slide 62 / 309 Measuring the Slope of a Line slope = rise run 10 8 The slope is the same anywhere on a line, so 6 run it can be measured 4 anywhere on the line. rise 2 0 -8 -2 -10 -6 -4 2 4 6 8 10 -2 -4 Keep in mind the direction: -6 · Up (+) Down (-) -8 · Right (+) Left (-) -10 Slide 63 / 309 Measuring the Slope of a Line For instance, in this case we measure the 10 slope by using a run from x = 0 to x = +6: a 8 run of 6. 6 rise 4 During that run, the line rises from y = 0 to 2 run y = 8: a rise of 8. slope = rise 0 -10 -8 -6 -4 -2 2 4 6 8 10 run -2 m = 8 -4 6 -6 m = 4 -8 3 -10
Slide 64 / 309 Measuring the Slope of a Line But we get the same result with a run from 10 x = 0 to x = +3: a run of 3. 8 6 During that run, the 4 line rises from y = 0 to y = 4: a rise of 4. 2 rise run slope = rise 0 -10 -8 -6 -4 -2 2 4 6 8 10 run -2 m = 4 -4 3 -6 -8 -10 Slide 65 / 309 Measuring the Slope of a Line But we can also start 10 at x = 3 and run to x = run 6 : a run of 3. 8 6 rise During that run, the 4 line rises from y = 3 to y = 7: a rise of 4. 2 0 -8 -4 -2 -10 -6 2 4 6 8 10 -2 slope = rise run -4 m = 4 -6 3 -8 -10 Slide 66 / 309 Measuring the Slope of a Line But we can also start at x = -6 and run to x = 10 0: a run of 6. 8 During that run, the 6 line rises from y = -8 to 4 y = 0: a rise of 8. 2 run slope = rise 0 -10 -8 -6 -4 -2 2 4 6 8 10 run -2 rise -4 m = 8 -6 6 m = 4 -8 3 -10
Slide 67 / 309 Measuring the Slope of a Line How is the slope different on this coordinate plane? 10 8 The line rises 8, however the run goes 6 left 6(negative). 4 Therefore, it is said to run 2 have a negative slope slope = rise 0 -10 -8 -6 -4 -2 2 4 6 8 10 rise run -2 -4 m = 8 -6 -6 m = -4 -8 3 -10 *most often the negative sign is placed in the numerator Slide 68 / 309 Slide 69 / 309
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Slide 76 / 309 Tables and Slope Return to Table of Contents Slide 77 / 309 How can slope and the y-intercept be found within the table? · Look for the change in the y-values · Look for the change in the x-values · Write as a ratio (simpified) - this will be the "slope" · Determine the corresponding y-value to the x-value of 0 - this will be the "y- intercept" x y -3 -1 0 5 3 11 Slide 78 / 309 x y -3 -1 +3 +6 0 5 +3 +6 3 11 6 = 2 is the slope 5 is the y-intercept 3
Slide 79 / 309 Determine the slope and y-intercept from this table. x y 5 -5 0 -4 -5 -3 is the slope -4 is the y-intercept click to reveal answer Slide 80 / 309 Slide 81 / 309
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Slide 85 / 309 Slope Formula Return to Table of Contents Slide 86 / 309 Slope is "the rise over the run"of a line. This idea of rise over run of a line on a graph is how we were able to determine the slope of a line. But slope can be found in other ways than looking at a graph. Slide 87 / 309 Slope is the ratio of change in y (rise) divided by the change in x(run). rise change in y slope= = run change in x A line has a constant ratio of change: A constant increase A constant decrease No change, just constant Or undefined slope
Slide 88 / 309 Another Application of the Definition of Slope Slope of a line is meant to measure how fast it is climbing or descending. A road might rise 1 foot for every 10 feet of horizontal distance. 1 foot 10 feet The ratio, 1/10, which is called slope, is a measure of the steepness of the hill. Engineers call this use of slope grade. What do you think a grade of 4% means? Slide 89 / 309 Slope of 3/20 3 feet 20 feet (The grade of this hill is slope of -3/7 3/20 = .15= 15%) 3 feet 7 feet (The grade of this hill is 3/7 = .43= 43%) Slide 90 / 309 so we will define the slope of a line as: (Rise) vertical change between two point on the line slope = horizontal change between two point on the line (Run)
Slide 91 / 309 Suppose point P = (x 1 , y 1 ) and Q = (x 2, y 2 ) are on the line whose slope we want to find. y Q(x 2 ,y 2 ) Vertical Change (y 2 -y 1 ) P(x 1 ,y 1 ) Horizontal Change (x 2 ,y 1 ) (x 2 -x 1 ) x The slope of line PQ= (y 2 -y 1 ) (x 2 -x 1 ) Slide 92 / 309 The vertical change between P and Q = y 2 - y 1 The horizontal change = x 2 - x 1 y 2 - y 1 slope = x 2 - x 1 Slide 93 / 309
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Slide 97 / 309 Slide 98 / 309 Slope-Intercept Form y = mx + b Return to Table of Contents Slide 99 / 309 Once you have identified the slope and y-intercept in an equation, it is easy to graph it! To graph y = 3x + 5...follow these steps: · Plot the y-intercept, in this case (0, 5) · Use the simplified rise over run to plot the next point - in this case, from (0, 5) go UP 3 units and RIGHT 1 unit to plot the next point. Connect the points.
Slide 100 / 309 Try this...graph y = -2x - 3 · Start at the y-intercept - plot it. · From the y-intercept, use the slope "m" to plot the next point. How would you use rise over run to plot -2? · Connect the points. click to reveal Did you have different points plotted? Does it make a difference? Slide 101 / 309 Try this...graph 4y = x + 12 (is this in y=mx + b form??) · Start at the y-intercept - plot it. · From the y-intercept, use the slope "m" to plot the next point. How would you use rise over run to plot it? · Connect the points. click to reveal Did you have different points plotted? Does it make a difference? Slide 102 / 309 Try this...graph 5x + y = -4 (is this in y=mx + b form??) · Start at the y-intercept - plot it. · From the y-intercept, use the slope "m" to plot the next point. How would you use rise over run to plot it? · Connect the points. click to reveal Did you have different points plotted? Does it make a difference?
Slide 103 / 309 Position of a Line 10 8 6 4 2 0 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 Slide 104 / 309 What are the similarities and differences between the lines below? 10 8 6 4 2 0 -8 -4 -2 -10 -6 2 4 6 8 10 -2 h(x)=x+6 -4 -6 q(x)=x+2 -8 -10 r(x)=x-1 s(x)=x-5 Slide 105 / 309 The lines were in the form of y = mx+b.
Slide 106 / 309 8 6 4 2 0 -8 -4 -2 -10 -6 2 4 6 8 10 -2 h(x)=x+6 -4 -6 q(x)=x+2 -8 -10 r(x)=x-1 s(x)=x-5 So it is the b in y = mx + b that is responsible for the position of the line. Slide 107 / 309 What determines slope? Examine the following equations: y = 2x + 1 y = 3x + 1 y = -1/2 x + 1 y = -x + 1 What do the equations have in common? What is different? Slide 108 / 309 y=-3x+1 10 y=x+1 8 6 4 2 y=1 0 -8 -4 -2 -10 -6 2 4 6 8 10 -2 -4 y=-1/2x+1 -6 -8 y=-7x+1 -10
Slide 109 / 309 Any equation of the form y = mx + b gives a line where b is the y intercept m is the slope Slide 110 / 309 Click for an interactive web site to see how the position of the line changes as you change the slope and the y-intercept. Slide 111 / 309
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Slide 118 / 309 Slide 119 / 309 Rate of Change Return to Table of Contents Slide 120 / 309 Slope formula can be used to find the constant of change in a "real world" problem. When traveling on the highway, drivers will set the cruise control and travel at a constant speed this means that the distance traveled is a constant Distance increase. (miles) (3,180) The graph at the right represents such a trip. The car passed mile-marker 60 at 1 hour and mile- (1,60) marker 180 at 3 hours. Find the slope of the line and what it represents. Time (hours) 180 miles-60 miles 120 miles 60 miles m= = = 3 hours-1 hours 2 hours hour So the slope of the line is 60 and the rate of change of the car is 60 miles per hour.
Slide 121 / 309 If a car passes mile-marker 100 in 2 hours and mile-marker 200 in 4 hours, how many miles per hour is the car traveling? The information above gives us the ordered pairs (2,100) and (4,200). Now find the rate of change. Slide 121 (Answer) / 309 If a car passes mile-marker 100 in 2 hours and mile-marker 200 in 4 hours, how many miles per hour is the car traveling? Answer The information above gives us the ordered pairs (2,100) and (4,200). Now find the rate of change. [This object is a pull tab] Slide 122 / 309
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Slide 129 / 309 Slide 130 / 309 61 Two different proportional relationships are represented by the equation and the table. Proportion A Proportion B y = 9x The rate of change in Proportion A is ______ ______ then the rate of change to Proportion B. A 1.5 E more B 2.5 F less C 25.5 From PARCC sample test D 43.5 Slide 130 (Answer) / 309 61 Two different proportional relationships are represented by the equation and the table. Proportion A Proportion B y = 9x Answer (B) 2.5 (F) less The rate of change in Proportion A is ______ ______ then the rate of change to Proportion B. A 1.5 E more [This object is a pull tab] B 2.5 F less C 25.5 From PARCC sample test D 43.5
Slide 131 / 309 62 A pool cleaning service drained a full pool. The following table shows the number of hours it drained and the amount of water remaining in the pool at that time. Students type their answers here Part A Plot the points that show the relationship between the number of hours elapsed and the number of gallons of water left in the pool. Select a place on the grid to plot each point. (Grid on next slide.) From PARCC sample test Slide 132 / 309 Slide 132 (Answer) / 309 Answer [This object is a pull tab]
Slide 133 / 309 63 Part B (continued from previous question) The data suggests a linear relationship between the number of hours the pool had been draining and the number of gallons of water remaining in the pool. Assuming the relationship is linear, what does the rate of change represent in the context of this relationship. A The number of gallons of water in the pool after 1 hour. B The number of hours it took to drain 1 gallon of water. C The number of gallons drained each hour. D The number of gallons of water in the pool when it is full. From PARCC sample test Slide 133 (Answer) / 309 63 Part B (continued from previous question) The data suggests a linear relationship between the number of hours the pool had been draining and the number of gallons of water remaining in the pool. Assuming the relationship is linear, what does the rate of change represent in the context of this relationship. Answer C A The number of gallons of water in the pool after 1 hour. B The number of hours it took to drain 1 gallon of water. C The number of gallons drained each hour. D The number of gallons of water in the pool when it is full. [This object is a pull tab] From PARCC sample test Slide 134 / 309 64 Part C (continued from previous question) What does the y-intercept of the linear function repressent in the context of this relationship? A The number of gallons in the pool after 1 hour. B The number of hours it took to drain 1 gallon of water. C The number of gallons drained each hour. D The number of gallons of water in the pool when it is full. From PARCC sample test
Slide 134 (Answer) / 309 64 Part C (continued from previous question) What does the y-intercept of the linear function repressent in the context of this relationship? A The number of gallons in the pool after 1 hour. Answer B The number of hours it took to drain 1 gallon of water. D C The number of gallons drained each hour. D The number of gallons of water in the pool when it is full. [This object is a pull tab] From PARCC sample test Slide 135 / 309 65 Part D (continued from previous question) Which equation describes this relationship between the time elapsed and the number of gallons of water remaining in the pool? A y = -600 x + 15,000 B y = -600 x + 13,2000 C y = -1,200 x + 13,200 D y = -1,200 x + 15,000 From PARCC sample test Slide 135 (Answer) / 309 65 Part D (continued from previous question) Which equation describes this relationship between the time elapsed and the number of gallons of water remaining in the pool? Answer A y = -600 x + 15,000 A B y = -600 x + 13,2000 C y = -1,200 x + 13,200 D y = -1,200 x + 15,000 [This object is a pull tab] From PARCC sample test
Slide 136 / 309 66 Eric planted a seedling in his garden and recorded its height each week. The equation shown can be used to estimate the height h , in inches, of the seedling after w , weeks since Eric planted the seedling. Part A: What does the slope of the graph of the equation represent? A The height in inches, of the seedling after w weeks. B The height in inches, of the seedling when Eric planted it. C The increases of height in inches, of the seedling each week. D The total increase in the height in inches, of the seedling after w weeks. From PARCC sample test Slide 136 (Answer) / 309 66 Eric planted a seedling in his garden and recorded its height each week. The equation shown can be used to estimate the height h , in inches, of the seedling after w , weeks since Eric planted the seedling. Part A: What does the slope of the graph of the equation Answer represent? C A The height in inches, of the seedling after w weeks. B The height in inches, of the seedling when Eric planted it. [This object is a pull tab] C The increases of height in inches, of the seedling each week. D The total increase in the height in inches, of the seedling after w weeks. From PARCC sample test Slide 137 / 309 67 Part B (continued from previous question) The equation estimates the height of the seedlings to be 8.25 inches after how many weeks? From PARCC sample test
Slide 137 (Answer) / 309 67 Part B (continued from previous question) The equation estimates the height of the seedlings to be 8.25 inches after how many weeks? Answer 8 [This object is a pull tab] From PARCC sample test Slide 138 / 309 Proportional Relationships Return to Table of Contents Slide 139 / 309 Pavers are being set around a birdbath. The figures below show the first three designs of the pattern. Using tiles, build the first five designs that follow the pattern above. Record your results in a table.
Slide 140 / 309 Design number 1 2 3 4 5 Number of 4 8 12 16 20 pavers Do the coordinate pairs in your table represent a proportional relationship? Graph the data from the table on a coordinate plane. What will you label the x-axis? What will you label the y-axis? Slide 141 / 309 Slide 142 / 309 How many tiles will you need for the "n-th" design? Write an equation that would represent the total number of tiles required for any design level. Suppose the birdbath was replaced with two tiles...how would this change the pattern? How would this change the equation?
Slide 142 (Answer) / 309 How many tiles will you need for the "n-th" design? Write an equation that would represent the total number of tiles required for any design t = 4n level. At each design level, the number of tiles would Answer increase by two after multiplying by four. Therefore Suppose the birdbath was replaced with two the equation would be: tiles...how would this change the pattern? How would this change the equation? t = 4n + 2 [This object is a pull tab] Slide 143 / 309 Graph both equations on the same coordinate plane. Discuss the similarities and differences in the graphs... Click for answer t=4n 15 15 Number of tiles t=4n+2 Number of tiles 10 10 5 5 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Design Level Design Level Slide 144 / 309 Slope & Similar Triangles Return to Table of Contents
Slide 145 / 309 Congruent triangles have the same shape and same size. Using the line as the hypotenuse, draw congruent right triangles. How do you know they are congruent? click to reveal example Slide 146 / 309 The vertical rise is the same as well as the horizontal run. The simplified ratio is the same as the absolute value of the slope. 4 2 4 2 4 2 1 2 4 4 2 2 Slide 147 / 309 Similar triangles have the same shape, however, they are not the same size. The corresponding sides are proportionate. 4 3 2 6 3 2 6 4
Slide 148 / 309 Sketch two similar right triangles on the line below. Write the ratios to prove they are proportionate. click to reveal example 4 8 2 1 Slide 149 / 309 Slide 150 / 309
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Slide 154 / 309 Slide 155 / 309 y Slide 156 / 309 75 Line t and ΔECA and ΔFDB are shown on the t coordinate grid. Which statements are true? Select all that apply. x A The slope of AC is equal to the slope of BC. B The slope of AC is equal to the slope of BD. C The slope of AC is equal to the slope of line t . D The slope of line t is equal to E The slope of line t is equal to F The slope of line t is equal to From PARCC sample test
y Slide 156 (Answer) / 309 75 Line t and ΔECA and ΔFDB are shown on the t coordinate grid. Which statements are true? Select all that apply. x A The slope of AC is equal to the slope of BC. B The slope of AC is equal to the slope of BD. Answer C The slope of AC is equal to A, B, C, E the slope of line t . D The slope of line t is equal to E The slope of line t is equal to [This object is a pull tab] F The slope of line t is equal to From PARCC sample test Slide 157 / 309 Complete the items below each table. (Click boxes to reveal answers) Family Z Family A Time (hr.) Distance (mi.) Time (hr.) Distance (mi.) from home from home 0 0 0 10 3 210 3 220 5 350 5 360 Slope (m) = 70 Slope (m) = 70 y-intercept (b) = 10 y-intercept (b) = 0 equation y = 70x + 10 equation y = 70x If this data from both tables were graphed on the same coordinate plane, what would you notice? Slide 158 / 309 Parallel and Perpendicular Lines Return to Table of Contents
Slide 159 / 309 The lines at the right are parallel lines. Notice that their slopes are all the same. 10 Parallel lines all have the slopes 8 because if they change at different 6 rates eventually they would 4 intersect. 2 0 -8 -4 -2 -10 -6 2 4 6 8 10 This also works for vertical and -2 h(x)=x+6 horizontal lines. -4 -6 q(x)=x+2 -8 r(x)=x-1 s(x)=x-5 -10 Slide 160 / 309 Slide 161 / 309
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Slide 165 / 309 Slide 166 / 309 Slide 167 / 309 In the diagram the 2 lines form a right angle, when this happens lines are said to perpendicular. h(x)=-3x-11 Look at their slopes. This time they g(x)= 1 / 3 x-2 are not the same instead they are opposite reciprocals
Slide 168 / 309 Perpendicular Equation Bank A) y=4x-2 is perpendicular to (Drag the equation to complete the statement.) B) y=- 1 / 5 x+1 is perpendicular to 6x+y=10 y= 1 / 5 x C) y-2=- 1 / 4 (x-3) is perpendicular to 1 / 5 y=x-2 D) 5x-y=8 is perpendicular to y=- 1 / 5 x+9 E) y= 1 / 6 x is perpendicular to y= 1 / 6 x-6 y=4x+1 F) y-9=-5(x-.4) is perpendicular to y=- 1 / 4 x-3 G) y=-6(x+2) is perpendicular to Slide 169 / 309 The rule of using opposite reciprocals will not work for Horizontal and Vertical Lines. Why? Discuss with your table. Slide 169 (Answer) / 309 The rule of using opposite reciprocals will not work for Horizontal and Vertical Lines. Why? Discuss with your table. Horizontal lines have a slope of zero. You can't take the Answer opposite reciprocal of 0. But the perpendicular line for a vertical line is a horizontal, and vice-versa. [This object is a pull tab]
Slide 170 / 309 Slide 171 / 309 Slide 172 / 309 Systems Strategy One: Graphing Return to Table of Contents
Slide 173 / 309 Some vocabulary... A "system" is two or more linear equations. The "solution" to a system is an ordered pair that will work in each equation. One way to find the solution is to graph the equations on the same coordinate plane and find the point of intersection. Slide 174 / 309 Consider this... Suppose you are walking to school. Your friend is 5 blocks ahead of you. You can walk two blocks per minute, your friend can walk one block per minute. How many minutes will it take for you to catch up with your friend? Slide 175 / 309 First, make a table to represent the problem. Friend's Your distance Time distance from from your start (min.) your start (blocks) (blocks) 0 5 0 1 6 2 2 7 4 3 8 6 4 9 8 5 10 10
Slide 176 / 309 Next, plot the points on a graph. 20 Friend's Your Time distance distance (min. from your from your 15 ) start start(blocks) (blocks) 0 5 0 Blocks 10 1 6 2 2 7 4 5 3 8 6 4 9 8 5 10 10 0 10 15 0 5 Time (min.) Slide 177 / 309 The point where they intersect is the solution to the system. 20 (5,10) is the solution. In the context of the 15 problem this means after 5 minutes, you Blocks will meet your friend 10 at block 10. 5 0 0 5 10 15 Time (min.) Slide 178 / 309 Solve the system of equations graphically. y = 2x -3 y = x - 1 Solution
Slide 179 / 309 Solve the system of equations graphically. 2x + y = 3 x - 2y = 4 Solution Slide 180 / 309 Solve the system of equations graphically. 3x + y = 11 x - 2y = 6 Solution Solve using graphing Slide 181 / 309 Write the equation for Write the y = -3x-1 y = 4x+6 move move the green equation for dashed line the blue solid line What is this point of intersection? (move the hand!) (-1, 2)
Slide 182 / 309 Now take the ordered pair we just found and substitute it into the equation to prove that it is a solution for both lines. ( , ) -1 2 y = -3x-1 y = 4x+6 Slide 183 / 309 Solve by Graphing y = 2x + 3 y = -4x - 3 Solution Slide 184 / 309 Solve by Graphing y= -3x + 4 y= x - 4 Solution
Slide 185 / 309 What's the problem here? y= 2x + 4 y= 2x - 4 Therefore Parallel there is no lines solution. do not intersect! No ordered ) ( pair that will work click to reveal in BOTH equations click to reveal Slide 186 / 309 Solve by Graphing First - transform the equations into y = mx + b form (slope-intercept form) 2x + y = 5 2y = -4x + 10 -2x -2x 2 2 y = -2x + 5 y = -2x + 5 Now graph the two transformed lines. Slide 187 / 309 What's the problem? 2x + y = 5 2y = 10 -4x becomes becomes y = -2x + 5 y = -2x + 5 The So we have equations infinitely transform to many the same solutions. line. click to reveal click to reveal
Slide 188 / 309 85 Solve the system by graphing. y = -x + 4 Solution y = 2x +1 A (3,1) Click for multiple choice answers. B (1,3) C (-1,3) no solution D Slide 189 / 309 86 Solve the system by graphing. y = 0.5x - 1 y = -0.5x -1 Solution A (0,-1) Click for multiple choice answers. B (0,0) C infinitely many D no solution Slide 190 / 309 87 Solve the system by graphing. 2x + y = 3 x - 2y = 4 Solution A (2,4) Click for multiple choice answers. B (0.4, 2.2) C (2, -1) D no solution
Slide 191 / 309 88 Solve the system by graphing. y = 3x + 3 Solution y = 3x - 3 A (0,0) Click for multiple choice answers. B (3,3) C infinitely many D no solution Slide 192 / 309 89 Solve the system by graphing. y = 3x + 4 4y = 12x + 16 A (3,4) Click for multiple choice answers. B (-3,-4) C infinitely many D no solution Slide 193 / 309 90 On the accompanying set of axes, graph and label the following lines: y=5 Solution x = - 4 y = x+5 Calculate the area, in square units, of the triangle formed by the three points of intersection. From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Slide 194 / 309 91 The equation of the line s is The equation of the line t is The equations of the lines s and t form a system of equations. Students type their answers here The solution of equations is located at Point P . From PARCC sample test Slide 194 (Answer) / 309 91 The equation of the line s is The equation of the line t is *Note: This question should be practiced on the computer in the PARCC sample test The equations of the lines s and t form a system of equations. so that students see how to graph the two lines on the computer. Students type their answers here The solution of equations is located at Point P . Answer [This object is a pull tab] From PARCC sample test Slide 195 / 309 92 The table shows two systems of linear equations. Indicate whether each system of the equations has no solution, one solution or infinitely many solutions by selecting the correct cell in the table. Select one cell per column. Students type their answers here From PARCC sample test
Slide 195 (Answer) / 309 92 The table shows two systems of linear equations. Indicate whether each system of the equations has no solution, one solution or infinitely many solutions by selecting the correct cell in the table. Select one cell per column. Answer Students type their answers here [This object is a pull tab] From PARCC sample test Slide 196 / 309 Systems Strategy Two: Substitution Return to Table of Contents Slide 197 / 309 Solve the system of equations graphically. y = x + 6.1 y = -2x - 1.4 NOTE
Slide 198 / 309 Substitution Explanation Graphing can be inefficient or approximate. Another way to solve a system is to use substitution. Substitution allows you to create a one variable equation. Slide 199 / 309 Solve the system using substitution. Why was it difficult to solve this system by graphing? y = x + 6.1 y = -2x - 1.4 y = -2x - 1.4 -start with one equation x + 6.1 = -2x - 1.4 -substitute x + 6.1 for y in equation +2x -6.1 +2x - 6.1 3x = -7.5 -solve for x x = -2.5 Substitute -2.5 for x in either equation and solve for y. y = x + 6.1 y = ( -2.5) + 6.1 y = 3.6 Since x = -2.5 and y = 3.6, the solution is (-2.5, 3.6) CHECK: See if (-2.5, 3.6) satisfies the other equation. y = -2x - 1.4 3.6 = -2(-2.5) - 1.4 ? 3.6 = 5 - 1.4 ? 3.6 = 3.6 Slide 200 / 309 Solve the system using substitution. (*Note: Equations can be moved on the page to show substitution into the y of the second equation.) y = -2x +14 ( ) -3 y + 3x = 21
Slide 200 (Answer) / 309 Solve the system using substitution. (*Note: Equations can be moved on the page to show substitution into the y of the second equation.) -3 (-2x + 14) + 3x = 21 6x - 42 + 3x = 21 y = -2x +14 ( ) 9x - 42 = 21 9x = 63 -3 y + 3x = 21 Answer x = 7 y = -2(7) + 14 y = -14 + 14 y = 0 (7, 0) [This object is a pull tab] Slide 201 / 309 Solve the system using substitution. ( ) x = -5y - 39 x = -y - 3 Slide 201 (Answer) / 309 Solve the system using substitution. -y - 3 = -5y - 39 x = -y -3 ( ) x = -5y - 39 4y - 3 = -39 x = -(-9) - 3 Answer 4y = -36 x = 9 -3 x y = -9 x = 6 = -y - 3 (6, -9) [This object is a pull tab]
Slide 202 / 309 Examine each system of equations. Which variable would you choose to substitute? Why? y = 4x - 9.6 y = -2x + 9 y = -3x 7x - y = 42 y = 4x + 1 x = 4y + 1 Slide 203 / 309 93 Examine the system of equations. Which variable would you substitute? 2x + y = 5 2y = 10 - 4x Solution A x B y Slide 204 / 309 94 Examine the system of equations. Which variable would you substitute? 2y - 8 = x Solution y + 2x = 4 A x B y
Slide 205 / 309 95 Examine the system of equations. Which variable would you substitute? x - y = 20 Solution 2x + 3y = 0 A x B y Slide 206 / 309 Sometimes you need to rewrite one of the equations so that you can use the substitution method. For example: The system: Is equivalent to: 3x -y = 5 y = 3x -5 2x + 5y = -8 2x + 5y = -8 Using substitution you now have: 2x + 5(3x-5) = -8 -solve for x 2x + 15x - 25 = -8 -distribute the 5 17x - 25 = -8 -combine x's 17x = 17 -at 25 to both sides x = 1 - divide by 17 Substitute x = 1 into one of the equations. 2(1) + 5y = -8 2 + 5y = -8 5y = -10 y = -2 The ordered pair (1,-2) satisfies both equations in the original system. 3x -y = 5 2x + 5y = -8 3(1) - (-2) = 5 2(1) + 5(-2) = -8 3 + 2 = 5 2 - 10 = -8 -8 = -8 Slide 207 / 309 Your class of 22 is going on a trip. There are four drivers and two types of vehicles, vans and cars. The vans seat six people, and the cars seat four people, including drivers. How many vans and cars does the class need for the trip? Let v = the number of vans and c = the number of cars
Slide 208 / 309 Set up the system: Drivers: v + c = 4 People: 6v + 4c = 22 Solve the system by substitution. v + c = 4 -solve the first equation for v. v = -c + 4 -substitute -c + 4 for v in the 6(-c + 4) + 4c = 22 second equation -6c + 24 + 4c = 22 -solve for c -2c + 24 = 22 -2c = -2 c = 1 v + c = 4 v + 1 = 4 -substitute for c in the 1st equation v = 3 -solve for v Since c = 1 and v = 3, they should use 1 car and 3 vans. Check the solution in the equations: v + c = 4 6v + 4c = 22 3 + 1 = 4 6(3) + 4(1) = 22 4 = 4 18 + 4 = 22 22 = 22 Slide 209 / 309 Now solve this system using substitution. What happens? x + y = 6 5x + 5y = 10 x + y = 6 -solve the first equation for x x = 6 - y 5(6 - y) + 5y = 10 -substitute 6 - y for x in 2nd equation 30 - 5y + 5y = 10 -solve for y 30 = 10 -FALSE! Since 30 = 10 is a false statement, the system has no solution. Slide 210 / 309 Now solve this system using substitution. What happens? x + 4y = -3 2x + 8y = -6 x + 4y = -3 - solve the first equation for x x = -3 - 4y 2(-3 - 4y) + 8y = -6 - sub. -3 - 4y for x in 2nd equation -6 - 8y + 8y = -6 - solve for y -6 = -6 - TRUE! - there are infinitely many solutions
Slide 211 / 309 How can you quickly decide the number of solutions a system has? 1 Solution Different slopes Same slope; different y- No Solution intercept (Parallel Lines) Same slope; same y-intercept Infinitely Many (Same Line) Slide 212 / 309 96 3x - y = -2 y = 3x + 2 Solution A 1 solution B no solution C infinitely many solutions Slide 213 / 309 97 3x + 3y = 8 1 y = x 3 Solution A 1 solution B no solution C infinitely many solutions
Slide 214 / 309 98 y = 4x 2x - 0.5y = 0 Solution A 1 solution B no solution C infinitely many solutions Slide 215 / 309 99 3x + y = 5 6x + 2y = 1 Solution A 1 solution B no solution C infinitely many solutions Slide 216 / 309 100 y = 2x - 7 y = 3x + 8 Solution A 1 solution B no solution C infinitely many solutions
Slide 217 / 309 101 Solve each system by substitution. y = x - 3 y = -x + 5 Solution Click for multiple choice answers. A (4,9) B (-4,-9) C (4,1) D (1,4) Slide 218 / 309 102 Solve each system by substitution. y = x - 6 y = -4 Solution Click for multiple choice answers. A (-10,-4) B (-4,2) C (2,-4) D (10,4) Slide 219 / 309 103 Solve each system by substitution. y + 2x = -14 y = 2x + 18 Solution Click for multiple choice answers. A (1,20) B (1,18) C (8,-2) D (-8,2)
Slide 220 / 309 104 Solve each system by substitution. 4x = -5y + 50 x = 2y - 7 Solution Click for multiple choice answers. A (6,6.5) B (5,6) C (4,5) D (6,5) Slide 221 / 309 105 Solve each system by substitution. y = -3x + 23 -y + 4x = 19 Solution Click for multiple choice answers. A (6,5) B (-7,5) C (42,-103) D (6,-5) Slide 222 / 309 Systems Strategy Three: Elimination Return to Table of Contents
Slide 223 / 309 When both linear equations of a system are in Standard Form, Ax + By = C, you can solve the system using elimination. You can add or subtract the equations to eliminate a variable. Slide 224 / 309 How do you decide which variable to eliminate? First, look to see if one variable has the same or opposite coefficients. If so, eliminate that variable. Second, look for which coefficients have a simple least common multiple. Eliminate that variable. Slide 225 / 309 If the variables have the same coefficient, you can subtract the two equations to eliminate the variable. If the variables have opposite coefficients, you add the two equations to eliminate the variable. Sometimes, you need to multiply one, or both, equations by a number in order to create a common coefficient.
Slide 226 / 309 Solve by Elimination - Click on the terms to eliminate and they will disappear, then add the two equations together. 5x + y = 44 ) ( -4x - y = -34 Slide 227 / 309 Solve by Elimination - Click on the terms and they will disappear then add the two equations together. 3x + y = 15 -3x -3y = -21 ( ) Slide 228 / 309 Solve by Elimination - There are 2 ways to complete this problem. See both examples. Multiplication by -1 5x + y = 17 5x + y = 17 -2x + y = -4 -2x + y = -4 Subtraction
Slide 229 / 309 Solve the system by elimination. 4x + 3y = 16 2x - 3y = 8 Pull Pull Slide 230 / 309 106 Solve each system by elimination. x + y = 6 x - y = 4 Solution Click for multiple choice answers. A (5,1) B (-5,-1) C (1,5) D no solution Slide 231 / 309 107 Solve each system by elimination. 2x + y = -5 2x - y = -3 Solution Click for multiple choice answers. A (-2,1) B (-1,-2) C (-2,-1) D infinitely many
Slide 232 / 309 108 Solve each system by elimination. 2x + y = -6 3x + y = -10 Solution Click for multiple choice answers. A (4,2) B (3,5) C (2,4) D (-4,2) Slide 233 / 309 109 Solve each system by elimination. 4x - y = 5 x - y = -7 Solution Click for multiple choice answers. A no solution B (4,11) C (-4,-11) D (11,-4) Slide 234 / 309 110 Solve each system by elimination. 3x + 6y = 48 -5x + 6y = 32 Solution Click for multiple choice answers. A (2,-7) B (7,2) C (2,7) infinitely many D
Slide 235 / 309 Sometimes, it is not possible to eliminate a variable by adding or subtracting the equations. When this is the case, you need to multiply one or both equations by a nonzero number in order to create a common coefficient. Then add or subtract the equations. Slide 236 / 309 Examine each system of equations. Which variable would you choose to eliminate? What do you need to multiply each equation by? 2x + 5y = -1 x + 2y = 0 3x + 8y = 81 5x - 6y = -39 3x + 6y = 6 2x - 3y = 4 Slide 237 / 309 In order to eliminate the y , you need to multiply first. 3x + 4y = -10 5x - 2y = 18 Multiply the second equation by 2 so the coefficients are opposites. 2(5x - 2y = 18) 10x - 4y = 36 Now solve by adding the equations together. 3x + 4y = -10 10x - 4y = 36 + 13x = 26 x = 2 Solve for y, by substituting x = 2 into one of the equations. 3x + 4y = -10 3(2) + 4y = -10 6 + 4y = -10 4y = -16 y = -4 So (2,-4) is the solution. Check: 3x + 4y = -10 5x - 2y = 18 3(2) + 4(-4) = -10 5(2) - 2(-4) = 18 6 + -16 = -10 10 + 8 = 18 -10 = -10 18 = 18
Slide 238 / 309 Now solve the same system by eliminating x . What do you multiply the two equations by? 3x + 4y = -10 5x - 2y = 18 Multiply the first equation by 5 and the second equation by 3 so the coefficients will be the same 5(3x + 4y = -10) 3(5x - 2y = 18) 15x + 20y = -50 15x - 6y = 54 Now solve by subtracting the equations. - 15x + 20y = -50 15x - 6y = 54 26y = -104 y = -4 Solve for x, by substituting y = -4 into one of the equations. 3x + 4y = -10 3x + 4(-4) = -10 3x + -16 = -10 3x = 6 x = 2 So (2,-4) is the solution. Check: 3x + 4y = -10 5x - 2y = 18 3(2) + 4(-4) = -10 5(2) - 2(-4) = 18 6 + -16 = -10 10 + 8 = 18 -10 = -10 18 = 18 Slide 239 / 309 111 Which variable can you eliminate with the least amount of work? Solution A x 9x + 6y = 15 -4x + y = 3 B y Slide 240 / 309 112 Which variable can you eliminate with the least amount of work? A x 3x - 7y = -2 Solution -6x + 15y = 9 B y
Slide 241 / 309 113 Which variable can you eliminate with the least amount of work? Solution A x x - 3y = -7 2x + 6y = 34 B y Slide 242 / 309 114 What will you multiply the first equation by in order to solve this system using elimination? 2x + 5y = 20 3x - 10y = 37 Now solve it.... Slide 242 (Answer) / 309 114 What will you multiply the first equation by in order to solve this system using elimination? You'd multiply 2x + 5y = 20 the first Answer 3x - 10y = 37 equation by 2. 2 - (11, ) 5 Now solve it.... [This object is a pull tab]
Slide 243 / 309 115 What will you multiply the first equation by in order to solve this system using elimination? 3x + 2y = -19 x - 12y = 19 Now solve it.... Slide 243 (Answer) / 309 115 What will you multiply the first equation by in order to solve this system using elimination? 3x + 2y = -19 You'd multiply Answer x - 12y = 19 the first equation by 6. (-5,-2) Now solve it.... [This object is a pull tab] Slide 244 / 309 116 What will you multiply the first equation by in order to solve this system using elimination? x + 3y = 4 3x + 4y = 2 Now solve it....
Slide 244 (Answer) / 309 116 What will you multiply the first equation by in order to solve this system using elimination? x + 3y = 4 You'd multiply 3x + 4y = 2 Answer the first equation by -3. (-2,2) Now solve it.... [This object is a pull tab] Slide 245 / 309 Systems Choose Your Strategy Return to Table of Contents Slide 246 / 309 Altogether 292 tickets were sold for a basketball game. An adult ticket costs $3. A student ticket costs $1. Ticket sales were $470. Let a = adults s = students
Slide 247 / 309 Set up the system: number of tickets sold: a + s = 292 money collected: 3a + s = 470 First eliminate one variable. a + s = 292 - in both equations s has the same - (3a + s = 470) coefficient so you subtract the 2 -2a+ 0 = -178 equations in order to eliminate it. a = 89 -solve for a Then, find the value of the eliminated variable. a + s = 292 89 + s = 292 -substitute 89 for a in 1st equation s = 203 -solve for s There were 89 adult tickets and 203 student tickets sold. (89, 203) Check: a + s = 292 3a + s = 470 89 + 203 = 292 3(89) + 203 = 470 292 = 292 267 + 203 = 470 470 = 470 Slide 248 / 309 117 A piece of glass with an initial temperature of 99 º F is cooled at a rate of 3.5 º F/min. At the same time, a piece of copper with an initial temperature of 0 º F is heated at a rate of 2.5º F/min. Let m = the number of minutes and t = the temperature in F. Which system Solution models the given information? A B C t = 99 + 3.5m t = 99 - 3.5m t = 99 + 3.5m t = 0 + 2.5m t = 0 + 2.5m t = 0 - 2.5m Slide 249 / 309 118 Which method would you use to solve the system? A graphing t = 99 - 3.5m B substitution t = 0 + 2.5m C elimination click for equations m = 16.5 Now solve it... t = 41.25 This means that in 16.5 minutes, the temperatures will both be 41.25º C. click for answer
Slide 249 (Answer) / 309 118 Which method would you use to solve the system? A graphing t = 99 - 3.5m B substitution t = 0 + 2.5m C elimination Answer click for equations B) Substitution m = 16.5 Now solve it... t = 41.25 [This object is a pull tab] This means that in 16.5 minutes, the temperatures will both be 41.25º C. click for answer Slide 250 / 309 119 What method would you choose to solve the system? 4s - 3t = 8 t = -2s -1 A graphing B substitution C elimination Slide 250 (Answer) / 309 119 What method would you choose to solve the system? 4s - 3t = 8 t = -2s -1 A graphing Answer B substitution B) Substitution C elimination [This object is a pull tab]
Slide 251 / 309 120 Now solve the system! Click for multiple choice answers. 1 ( , -2) A 4s - 3t = 8 2 t = -2s -1 1 B ( , 2) 2 C (2 , -2) 1 D (-2, ) 2 Slide 251 (Answer) / 309 120 Now solve the system! Click for multiple choice answers. 1 A ( , -2) 4s - 3t = 8 2 t = -2s -1 Answer 1 B ( , 2) A 2 C (2 , -2) [This object is a pull tab] 1 D (-2, ) 2 Slide 252 / 309 121 What method would you choose to solve the system? y = 3x - 1 A graphing y = 4x B substitution C elimination
Slide 252 (Answer) / 309 121 What method would you choose to solve the system? y = 3x - 1 A graphing y = 4x Answer B substitution B) substitution C elimination [This object is a pull tab] Slide 253 / 309 122 Now solve it! Click for multiple choice answers. A (1, 4) y = 3x - 1 y = 4x B (-4, -1) C (-1, 4) (-1, -4) D Slide 253 (Answer) / 309 122 Now solve it! Click for multiple choice answers. A (1, 4) y = 3x - 1 Answer y = 4x B (-4, -1) D (-1, 4) C (-1, -4) D [This object is a pull tab]
Slide 254 / 309 123 What method would you choose to solve the system? 3m - 4n = 1 A graphing 3m - 2n = -1 B substitution C elimination Slide 254 (Answer) / 309 123 What method would you choose to solve the system? 3m - 4n = 1 A graphing 3m - 2n = -1 Answer B substitution C) elimination C elimination [This object is a pull tab] Slide 255 / 309 124 Now solve it! Click for multiple choice answers. A (-2, -1) 3m - 4n = 1 3m - 2n = -1 B (-1, -1) C (-1, 1) D (1, 1)
Slide 255 (Answer) / 309 124 Now solve it! Click for multiple choice answers. A (-2, -1) 3m - 4n = 1 Answer 3m - 2n = -1 B (-1, -1) B C (-1, 1) D (1, 1) [This object is a pull tab] Slide 256 / 309 125 What method would you choose to solve the system? y = -2x A graphing y = -0.5x + 3 B substitution C elimination Slide 256 (Answer) / 309 125 What method would you choose to solve the system? y = -2x A graphing Answer y = -0.5x + 3 B substitution B C elimination [This object is a pull tab]
Slide 257 / 309 126 Now solve it! Click for multiple choice answers. A (-6, 12) y = -2x y = -0.5x + 3 B (2, -4) C (-2, 4) D (1, -2) Slide 257 (Answer) / 309 126 Now solve it! Click for multiple choice answers. A (-6, 12) y = -2x y = -0.5x + 3 B (2, -4) Answer C C (-2, 4) D (1, -2) [This object is a pull tab] Slide 258 / 309 127 What method would you choose to solve the system? A graphing 2x - y = 4 x + 3y = 16 B substitution C elimination
Slide 258 (Answer) / 309 127 What method would you choose to solve the system? A graphing 2x - y = 4 Answer x + 3y = 16 B substitution C C elimination [This object is a pull tab] Slide 259 / 309 128 Now solve it! Click for multiple choice answers. A (6, 5) 2x - y = 4 x + 3y = 16 B (-4, 7) C (-4, 4) D (4, 4) Slide 259 (Answer) / 309 128 Now solve it! Click for multiple choice answers. A (6, 5) 2x - y = 4 Answer x + 3y = 16 B (-4, 7) D C (-4, 4) D (4, 4) [This object is a pull tab]
Slide 260 / 309 129 What method would you choose to solve the system? u = 4v A graphing 3u - 3v = 7 B substitution C elimination Slide 260 (Answer) / 309 129 What method would you choose to solve the system? u = 4v A graphing 3u - 3v = 7 Answer B B substitution C elimination [This object is a pull tab] Slide 261 / 309 130 Now solve it! Click for multiple choice answers. A ( , ) 7 28 u = 4v 9 9 3u - 3v = 7 B ( , ) 7 28 9 9 C (28, 7) D 7 (7, ) 4
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