Solving Systems A system of linear equations is comprised of 2 or - PDF document

Slide 1 / 176 Slide 2 / 176 Algebra I System of Linear Equations 2015-11-02 www.njctl.org Slide 3 / 176 Slide 3 (Answer) / 176 Table of Contents Table of Contents Click on the topic to go to that section Click on the topic to go to that section Many of the Responder Solving Systems by Graphing Solving Systems by Graphing Questions have boxes over Teacher Note the answers. Have the Solving Systems by Substitution Solving Systems by Substitution students take some time to Solving Systems by Elimination Solving Systems by Elimination solve the system prior to showing them the multiple Choosing your Strategy Choosing your Strategy choice answers by clicking on Writing Systems to Model Situations Writing Systems to Model Situations the box, so they do not just substitute in the answers. Standards Standards [This object is a pull tab] Slide 4 / 176 Slide 5 / 176 System of Equations Solving Systems A system of linear equations is comprised of 2 or more linear equations. by The solution of the system will be the values of the Graphing variables which make all the equations true. Return to Table of Contents

Slide 6 / 176 Slide 7 / 176 Solve by Graphing Solve by Graphing Example: Example: y y y = 2x - 4 10 y = –x + 5 10 The graph of the line that represents the solutions Similarly, this graph is of 5 5 to the above equation is the line that represents shown. the solutions to this equation. x x It represents all the 0 0 -5 10 -10 5 -10 -5 5 10 points whose x and y It represents all the values make the above points whose x and y -5 equation true. -5 values make the above equation true. The line is easy to find -10 from the equation since -10 the equation is in slope- intercept form. Slide 8 / 176 Slide 9 / 176 Solve by Graphing Solve by Graphing Example: y Example: y y = 2x - 4 y = 2x - 4 10 10 y = –x + 5 y = –x + 5 5 Here are the lines that 5 At the point they represent the solutions cross, both equations to both those equations. must be true, since x Each line shows the x that point is on both 0 -10 -5 5 10 0 infinite set of solutions -10 -5 5 10 lines. for each equation. -5 They appear to cross -5 What must be true at (3, 2). Let's check about the point at which that in both equations. they cross? -10 -10 DISCUSS. Slide 10 / 176 Slide 11 / 176 Solve by Graphing System of Equations Example: y Not all systems have solutions...and some have an infinite number of solutions. Substitute x = 3 and 10 y = 2 into both equations Let's see how to figure out whether there are solutions, and see if both equations how many, and what they are. are true. 5 y = 2x - 4 x (2) = 2(3) - 4 0 -5 10 -10 5 2 = 2 correct Click here to watch a music video -5 that introduces what we will learn about systems. y = –x + 5 -10 (2) = -(3) + 5 2 = 2 correct

Slide 12 / 176 Slide 13 / 176 The Number of Solutions The Number of Solutions When graphing two lines there are three possibilities. So, systems of equations can have either: They meet in one point: the point of intersection. · 1 solution, if the lines meet at one point · They never meet: they are parallel. 0 solutions, if they never meet · · They meet at all their points: they are the same line. · Infinite solutions, if they are the same line · Slide 14 / 176 Slide 15 / 176 Type 1: One Solution Type 1: One Solution y y The two lines intersect in y = 2x - 4 10 exactly ONE place. 10 y = –x + 5 The solution is the point at 5 5 which they intersect. This is the example we started with. The slopes of the x x lines must be different, or As we confirmed there 0 -5 10 0 -10 5 -5 10 -10 5 they would never cross. is one solution to this system of equations: -5 (3, 2). -5 -10 -10 Slide 16 / 176 Slide 17 / 176 Type 2: No Solution Type 2: No Solution y = 2x + 6 y = 2x + 2 y y The lines never meet. Both are written in slope intercept form 10 10 There is no solution true for both lines. y = mx + b 5 5 to make it easy to The lines are parallel. compare slopes and y-intercepts. They must have the x x 0 0 same slope, since they -10 -5 5 10 -10 -5 5 10 The slope for both lines are parallel. is 2 (the coefficient of x). -5 -5 But, they must have So, the lines are parallel. different intercepts, or they would be the same -10 -10 The y-intercepts are line. different, +6 and +2, so the lines never cross.

Slide 18 / 176 Slide 19 / 176 Type 3: Infinite Solutions Type 3: Infinite Solutions y = 2x + 2 y = 2x + 2 y y The lines overlap at all Both are written in slope points. 10 10 intercept form They are different y = mx + b equations for the same 5 5 to make it easy to line. compare slopes and y- intercepts. The lines are parallel. x x 0 0 -5 10 -10 5 -10 -5 5 10 The slope for both lines is So, they must have the 2 (the coefficient of x). same slopes. -5 -5 So, the lines are parallel. The intercepts are the The y-intercepts for both same, since all their -10 -10 lines are +2, so the lines points are the same. overlap everywhere. Slide 20 / 176 Slide 21 / 176 The Number of Solutions Type 3: Infinite Solutions y = 2x + 2 y = 2x + 2 First, put the equations into slope-intercept form by solving for y. y In slope intercept form, the Then, decide on the number of solutions. fact that these are the same 10 line is obvious. After that, solutions can be found in three different ways. But, if the equations were 5 written as below, it would be less obvious: x 2y - 4x = 4 0 -10 -5 5 10 -6x = -3y + 6 -5 That's why it's always a good idea to put equations into slope-intercept -10 form...they're easier to read, graph and compare. Slide 22 / 176 Slide 23 / 176 How can you quickly decide the Solving both Equations for y number of solutions a system has? Let's solve this system of equations y = -5x + 4 10x + 2y = 6 Different slopes 1 Solution Math Practice Different lines The equation on the left is in slope-intercept form. Same slope Do you see that the slope is -5 and its y-intercept is +4? No Solution Different y -intercept Parallel Lines The equation on the right is not in slope-intercept form, so we can't Same slope see it's slope or y-intercept. Infinitely Many Same y -intercept So, we can't tell yet how many solutions will satisfy both equations. Same Line Let's solve the second equation for y.

Slide 24 / 176 Slide 25 / 176 Solving for y Solving both Equations for y 10x + 2y = 6 y= -5x + 4 Original Equations 10x + 2y = 6 Subtract 10x from both sides -10x -10x 2y = -10x + 6 y= -5x + 4 Slope Intercept Form y = -5x + 3 Divide both sides by 2 2y = -10x + 6 m = -5 b = 4 Slopes and Intercepts m = -5 b = 3 2 2 The slopes are the same but the y-intercepts are different. y = -5x + 3 This is now in slope-intercept form. How many solutions are there? We can see the slope and y-intercept m = -5 b = 3 Slide 26 / 176 Slide 27 / 176 Solving both Equations for y Solving for y Let's solve this system of equations 6x + 2y = 4 Subtract 6x from both sides y = 2x + 5 6x + 2y = 4 -6x -6x The equation on the left is in slope-intercept form and we can 2y = -6x + 4 see the slope is +2 and the y-intercept is +5. Divide both sides by 2 The equation on the right is not in slope-intercept form, let's solve that equation for y. 2y = -6x + 4 2 2 y = -3x + 2 This is now in slope-intercept form. m = -3 b = +2 Slide 28 / 176 Slide 29 / 176 Solving both Equations for y 1 How many solutions does this system have: y = 2x - 7 y= -5x + 4 Original Equations 6x + 2y = 4 y = 3x + 8 y= -5x + 4 Slope Intercept Form y = -3x + 2 A 1 solution m = -5 b = 4 Slopes and Intercepts m = -3 b = +2 B no solution C infinitely many The slopes are different. solutions How many solutions are there?

Slide 29 (Answer) / 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 Answer A A 1 solution A 1 solution B no solution B no solution C infinitely many [This object is a pull tab] solutions C infinitely many solutions Slide 30 (Answer) / 176 Slide 31 / 176 2 How many solutions does this system have: 3 How many solutions does this system have: 3x + 3y = 8 3x - y = -2 1 Answer y = x y = 3x + 2 3 C A 1 solution A 1 solution [This object is a pull tab] B no solution B no solution C infinitely many C infinitely many solutions solutions Slide 31 (Answer) / 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 Answer 3 A A 1 solution A 1 solution B no solution B no solution [This object is a pull tab] C infinitely many C infinitely many solutions solutions