Solving Systems Solving Systems by Graphing by Solving Systems by - PDF document

Slide 1 / 176 Slide 2 / 176 Algebra I System of Linear Equations 2015-11-02 www.njctl.org Slide 3 / 176 Slide 4 / 176 Table of Contents Click on the topic to go to that section Solving Systems Solving Systems by Graphing by Solving Systems by Substitution Graphing Solving Systems by Elimination Choosing your Strategy Writing Systems to Model Situations Return to Table Standards of Contents Slide 5 / 176 Slide 6 / 176 System of Equations Solve by Graphing Example: y A system of linear equations is comprised of 2 or more linear equations. y = 2x - 4 10 The solution of the system will be the values of the The graph of the line that variables which make all the equations true. represents the solutions 5 to the above equation is shown. x It represents all the 0 -5 10 -10 5 points whose x and y values make the above equation true. -5 The line is easy to find from the equation since -10 the equation is in slope- intercept form.

Slide 7 / 176 Slide 8 / 176 Solve by Graphing Solve by Graphing Example: Example: y y y = 2x - 4 10 10 y = –x + 5 y = –x + 5 Similarly, this graph is of 5 5 Here are the lines that the line that represents represent the solutions the solutions to this to both those equations. equation. x x Each line shows the 0 0 -5 10 -10 -5 5 10 -10 5 infinite set of solutions It represents all the for each equation. points whose x and y -5 -5 values make the above What must be true equation true. about the point at which they cross? -10 -10 DISCUSS. Slide 9 / 176 Slide 10 / 176 Solve by Graphing Solve by Graphing Example: Example: y y Substitute x = 3 and y = 2x - 4 10 10 y = 2 into both equations y = –x + 5 and see if both equations are true. 5 5 At the point they cross, both equations y = 2x - 4 must be true, since x x that point is on both (2) = 2(3) - 4 0 0 -10 -5 5 10 -10 -5 5 10 lines. 2 = 2 correct -5 They appear to cross -5 at (3, 2). Let's check that in both equations. y = –x + 5 -10 -10 (2) = -(3) + 5 2 = 2 correct Slide 11 / 176 Slide 12 / 176 System of Equations The Number of Solutions When graphing two lines there are three possibilities. Not all systems have solutions...and some have an infinite number of solutions. They meet in one point: the point of intersection. · Let's see how to figure out whether there are solutions, They never meet: they are parallel. · how many, and what they are. They meet at all their points: they are the same line. · Click here to watch a music video that introduces what we will learn about systems.

Slide 13 / 176 Slide 14 / 176 The Number of Solutions Type 1: One Solution y So, systems of equations can have either: The two lines intersect in 10 1 solution, if the lines meet at one point exactly ONE place. · 0 solutions, if they never meet · The solution is the point at 5 which they intersect. Infinite solutions, if they are the same line · The slopes of the x lines must be different, or 0 -10 -5 5 10 they would never cross. -5 -10 Slide 15 / 176 Slide 16 / 176 Type 2: No Solution Type 1: One Solution y y The lines never meet. 10 y = 2x - 4 10 There is no solution true for both lines. y = –x + 5 5 5 This is the example we The lines are parallel. started with. They must have the x x 0 As we confirmed there same slope, since they -10 -5 5 10 0 -5 10 -10 5 is one solution to this are parallel. system of equations: -5 (3, 2). -5 But, they must have different intercepts, or they would be the same -10 line. -10 Slide 17 / 176 Slide 18 / 176 Type 2: No Solution Type 3: Infinite Solutions y = 2x + 6 y = 2x + 2 y y Both are written in slope The lines overlap at all intercept form 10 points. 10 y = mx + b They are different 5 equations for the same to make it easy to 5 line. compare slopes and y-intercepts. x The lines are parallel. 0 x -10 -5 5 10 The slope for both lines 0 -10 -5 5 10 is 2 (the coefficient of x). So, they must have the -5 same slopes. -5 So, the lines are parallel. The intercepts are the -10 same, since all their The y-intercepts are -10 different, +6 and +2, so points are the same. the lines never cross.

Slide 19 / 176 Slide 20 / 176 Type 3: Infinite Solutions Type 3: Infinite Solutions y = 2x + 2 y = 2x + 2 y = 2x + 2 y = 2x + 2 y y In slope intercept form, the Both are written in slope 10 fact that these are the same 10 intercept form line is obvious. y = mx + b But, if the equations were 5 5 to make it easy to written as below, it would compare slopes and y- be less obvious: intercepts. x x 0 2y - 4x = 4 0 -5 10 -10 5 -10 -5 5 10 The slope for both lines is -6x = -3y + 6 2 (the coefficient of x). -5 -5 That's why it's always a So, the lines are parallel. good idea to put equations The y-intercepts for both into slope-intercept -10 -10 lines are +2, so the lines form...they're easier to read, graph and compare. overlap everywhere. Slide 21 / 176 Slide 22 / 176 The Number of Solutions How can you quickly decide the number of solutions a system has? First, put the equations into slope-intercept form by solving for y. Then, decide on the number of solutions. Different slopes 1 Solution Math Practice Different lines After that, solutions can be found in three different ways. Same slope No Solution Different y -intercept Parallel Lines Same slope Infinitely Many Same y -intercept Same Line Slide 23 / 176 Slide 24 / 176 Solving both Equations for y Solving for y Let's solve this system of equations 10x + 2y = 6 Subtract 10x from both sides y = -5x + 4 10x + 2y = 6 -10x -10x The equation on the left is in slope-intercept form. 2y = -10x + 6 Divide both sides by 2 Do you see that the slope is -5 and its y-intercept is +4? 2y = -10x + 6 2 2 The equation on the right is not in slope-intercept form, so we can't see it's slope or y-intercept. y = -5x + 3 This is now in slope-intercept form. So, we can't tell yet how many solutions will satisfy both equations. We can see the slope and y-intercept Let's solve the second equation for y. m = -5 b = 3

Slide 25 / 176 Slide 26 / 176 Solving both Equations for y Solving both Equations for y Let's solve this system of equations y= -5x + 4 Original Equations 10x + 2y = 6 y = 2x + 5 6x + 2y = 4 The equation on the left is in slope-intercept form and we can y= -5x + 4 Slope Intercept Form y = -5x + 3 see the slope is +2 and the y-intercept is +5. The equation on the right is not in slope-intercept form, let's m = -5 b = 4 Slopes and Intercepts m = -5 b = 3 solve that equation for y. The slopes are the same but the y-intercepts are different. How many solutions are there? Slide 27 / 176 Slide 28 / 176 Solving for y Solving both Equations for y y= -5x + 4 Original Equations 6x + 2y = 4 6x + 2y = 4 Subtract 6x from both sides -6x -6x y= -5x + 4 Slope Intercept Form y = -3x + 2 2y = -6x + 4 Divide both sides by 2 m = -5 b = 4 Slopes and Intercepts m = -3 b = +2 2y = -6x + 4 2 2 The slopes are different. y = -3x + 2 This is now in slope-intercept form. How many solutions are there? m = -3 b = +2 Slide 29 / 176 Slide 30 / 176 2 How many solutions does this system have: 1 How many solutions does this system have: y = 2x - 7 3x - y = -2 y = 3x + 8 y = 3x + 2 A 1 solution A 1 solution B no solution B no solution C infinitely many solutions C infinitely many solutions

Slide 31 / 176 Slide 32 / 176 3 How many solutions does this system have: 4 How many solutions does this system have: y = 4x 3x + 3y = 8 1 2x - 0.5y = 0 y = x 3 A 1 solution A 1 solution B no solution B no solution C infinitely many C infinitely many solutions solutions Slide 33 / 176 Slide 34 / 176 5 How many solutions does this system have: Consider this... 3x + y = 5 Suppose you are walking to school. Your friend is 5 blocks ahead of you. You can walk two blocks per 6x + 2y = 1 minute and your friend can walk one block per minute. How many minutes will it take for you to catch up with your friend? A 1 solution B no solution C infinitely many solutions Slide 35 / 176 Slide 36 / 176 Solution Solution Continued First, make a table to represent the problem. Next, plot the points on a graph. Blocks Friend's Friend's Your Your distance distance Time distance from distance from your start Time from (min.) your start from your 10 (blocks) (min.) your (blocks) start start (blocks) (blocks) 0 5 0 0 5 0 1 6 2 5 1 6 2 2 7 4 2 7 4 3 8 6 3 8 6 4 9 8 4 9 8 0 5 10 5 10 10 5 10 10 Time (min.)