`NameCHAPTERDateClassReteach11-3 Solving Equations with Variables on Both SidesSome equations have like terms that are on opposite sides of the equal sign. To solve these equations, first identify the like term with the lesser coefficient. 2x 25 3x 2x and 3x are like terms. Since 3 2, 3x is the term with the 2x lesser coefficient. Move 3x to the 2x 3x opposite side of the equation by adding 5x 5x 3x to each side of the equation.525 25 25 525 53x 3x3xxComplete the steps to solve the equation. 1. 5x 16 9x . 2. 3a 1 15 4a .The like terms are and The like term with the lesser coefficient is . 9x 5x 16 5x 16 9x 1616 4xThe like terms are and The like term with the lesser coefficient is . 3a 1 15 4a 3a 1 15 4a 1 15 7a 1 7a7a 1415xa Solve. 3. 8d 7d 13 4. 4p 16 4p 5. 3n 25 2n6. 9c62c157. 166k202k8. 2h82hCopyright © by Holt, Rinehart and Winston. All rights reserved.30Holt Middle School Math Course 2NameLESSONDateClassPractice B11-3 Solving Equations with Variables on Both SidesGroup the terms with the variables on one side of the equal sign and simplify. Do not solve. 1. 10t 6t 24 2. 6x 32 2x 3. j 20 4j4.5d405d5. 9m282m6. 9 x884 x 9Solve. 7. 8k 6k 26 8. 7p 25 12p 9. 3r 18 6r10. 8w93w1611. 325v3v812.12y106y1413. 8 a563 a 414. 4 n1102 n 315. 201 d 53 d 1016Copyright © by Holt, Rinehart and Winston. All rights reserved.28Holt Middle School Math Course 2NameLESSONDateClassReteachyrise 4 2 4 2 2 right (4, 4)run12-5 SlopeThe slope of a line is a ratio that measures the steepness of that line. The sign of the slope tells whether the line is rising or falling. You can find the slope of a line by comparing any two points on that line. Find the slope of the line between (2, 1) and (4, 4). sloperise run up ( ) or down ( ) right ( ) or left ( )4 4 1 2 3 23 up 2(2, 1) 4xO2 4Use the graph to complete the statements. 1. You have to move up sloperise runand rightto go from one point to the other.3 2up ( ) or down ( ) right ( ) or left ( )2. The slope of the line is.You can find the y-intercept of a line by locating the point where it crosses the y-axis. The y-intercept is the y-coordinate of that point. 3. The line crosses the y-axis at the point ( ). The y-intercept is .Tell the slope and y-intercept of each line given its graph. 4.4 2 (0, 1) 4 2y5.4 2yx2 (1, 4 2) 4 2 ( 2,x O2) 4 (2, 2 2) 4O2 4slopeup or down right or leftslopeup or down right or lefty-interceptCopyright © by Holt, Rinehart and Winston. All rights reserved.y-intercept 48Holt Middle School Math Course 2NameLESSONDateClassReteach12-5 Slope (continued)You can find the slope and y-intercept of a line from the equation when the equation is in the form y mx b. This is called slope-intercept form. The variable m represents the slope, and b represents the y-intercept. y y mx 4x b 22y ymx2 x 3bThe slope is 4, and the y-intercept is 2.The slope is 3 , and the y-intercept is 0.Tell the slope and the y-intercept of each line, given the equation. 6. y 5x 4 7. y1 x 218. y3 x 2Using the slope and the y-intercept, you can graph the equation y3 x 4y3.4 2 4 2 3 up 2 4 4 right (4, 0) x O 2 4 (0, 3)The y-intercept is 3, so the graph contains the point (0, 3). The slope is 4 . Starting at (0, 3), count up 3, then 4 to the right. This is the point (4, 0). Then draw a line through the two points.39. Graph y 3x and y-intercept. y-intercept slope5 using the slopey; plot point (1)4 2From the y-intercept, count down and right . Plot that point and drawx4 2O224a line through the two points.Copyright © by Holt, Rinehart and Winston. All rights reserved.49Holt Middle School Math Course 2NameLESSONDateClassPractice B12-5 SlopeTell the slope and y-intercept of each line, given its graph. 1.4 2 4 2y(1, 3) (2, 2) (3, 1)2.4 2y(3, 2)x O2 4 2 4 4 2x O2 (0, 4 2 2) 4Tell the slope and y-intercept, given the equation. 3. y 2x 3 4. y1 x 355. y4x6. y4x17. yx48. y2x8Graph each line, given the equation. 9. y1 x 25y10. y4y44 2 2x x4 2O2 42442O224Copyright © by Holt, Rinehart and Winston. All rights reserved.46Holt Middle School Math Course 2NameLESSONDateClassReteachThe graph of a nonlinear function is not a straight line. The slope of a nonlinear function changes as the line curves.y6 412-6 Nonlinear FunctionsThe graph of a linear function is a straight line. The slope of the line is constant.y4 2x4 2O2 424 4 22x O2 2 4Tell whether the graph is linear or nonlinear. 1.6 4 2 4 2y2.4 2yx O2 4x4 224O224Is the graph a straight line? The graph is .Is the graph a straight line? The graph is .Copyright © by Holt, Rinehart and Winston. All rights reserved.59Holt Middle School Math Course 2NameLESSONDateClassPractice A12-6 Nonlinear FunctionsTell whether the graph is linear or nonlinear. 1. This is the graph of yy4 2 4 23x3.2. This is the graph of yyx 2.x4 2x4 2O2 424O2 424Is the graph a straight line?Is the graph a straight line?3. This is the graph of yy4 2x21.4. This is the graph of yy4 2x.x4 2x4 2O2 424O2 424Is the graph a straight line?Is the graph a straight line?Tell whether the function in each table could be linear or is nonlinear. 5. x 5 6 7 y 25 36 49 6. x 2 4 6 y 100 300 500Copyright © by Holt, Rinehart and Winston. All rights reserved.56Holt Middle School Math Course 2`

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