Page 1 of 1-7 Equations Teks Focus TEKS (2)(B) Derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines. TEKS (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate. Additional TEKS (1)(E) Vocabulary Point-slope form The point-slope form for a nonvertical line with slope m and through point (x 1, y 1) is y y 1 = m(x x 1). Slope the ratio of a line's vertical change (rise) to its corresponding horizontal change (run) between any two points in the coordinate plane Slope-intercept form The slope-intercept form of the equation of a nonvertical line is y = mx + b, where m is the slope and b is the y-intercept. Implication a conclusion that follows from previously stated ideas or reasoning without being explicitly stated Representation a way to display or describe information. You can use a representation to present mathematical ideas and data. ESSENTIAL UNDERSTANDING You can graph a line and write its equation when you know certain facts about the line, such as its slope and a point on the line. take note Key Concept Slope Definition Symbols Diagram The slope m of a line is the ratio of the vertical change (rise) to the horizontal change (run) between any two points. A line contains the points (x 1, y 1) and (x 2, y 2). rise y m = run = 2 y 1 x 2 x 1 The slope of a line can be positive, negative, zero, or undefined. The sign of the slope tells you whether the line rises or falls to the right. A slope of zero tells you that the line is horizontal. An undefined slope tells you that the line is vertical. PearsonTEXAS.com Page 12 Copyright 2016 Pearson Education, Inc. or its affiliate(s). All rights reserved. Privacy Policy Terms of Use Rights and Permissions

Page 1 of 2 take note Key Concept Forms of Linear Equations Definition Symbols The slope-intercept form of an equation of a nonvertical line is y = mx + b, where m is the slope and b is the y-intercept. The point-slope form of an equation of a nonvertical line is y y 1 = m(x x 1), where m is the slope and (x 1, y 1) is a point on the line. Problem 1 TEKS Process Standard (1)(D) Deriving the Slope Formula Derive a formula that you can use to find the slope of a line. The slope of a line is the ratio of the vertical change to the horizontal change between any two points. Step 1 Mark points (x 1, y 1) and (x 2, y 2), and draw a line through them. Step 2 Since the y-coordinates correspond to values on a vertical number line (y-axis), use the definition of distance between two points to find the vertical change. The vertical change is the difference between y 2 and y 1, or y 2 y 1. Step Since the x-coordinates correspond to values on a horizontal number line (x-axis), use the definition of distance between two points to find the horizontal change. The horizontal change is the difference between x 2 and x 1, or x 2 x 1. Step 4 Find the slope m by determining the ratio of vertical change to horizontal change. So the slope formula is m = y 2 y 1. x 2 x 1 Problem 2 Think What reason can you give for drawing a line through points (x 1, y 1) and (x 2, y 2) in Step 1? Postulate 1-1 states that through any two points, there is exactly one line. Finding Slopes of Lines A What is the slope of line b? m = 2 ( 2) 1 4 = 4 5 = 4 5 B What is the slope of line d? m = 0 ( 2) 4 4 = 2 0 Undefined Plan How do you know which numbers go where in a formula? For slope, you can choose either point's y-coordinate as y 2. Just be sure to use the same point's x-coordinate as x 2.

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Page 1 of 1 Problem TEKS Process Standard (1)(E) Graphing Lines 2 A What is the graph of y = x +1? The equation is in slope-intercept form, y = mx + b. The slope m is, and the y-intercept b is 1. 2 Plan What do you do first? Determine which form of linear equation you have. Then use the equation to identify the slope and a starting point. d B What is the graph of y = 2(x + )? The equation is in point-slope form, y y 1 = m(x x 1). The slope m is 2, and a point (x 1, y 1) on the line is (, ). d Think What do you do when the slope is not a fraction? A rise always needs a run. So write the integer as a fraction with 1 as the denominator. PearsonTEXAS.com Page 125 Copyright 2016 Pearson Education, Inc. or its affiliate(s). All rights reserved. Privacy Policy Terms of Use Rights and Permissions

Page 1 of 2 Problem 4 Writing Equations of Lines A What is an equation of the line with slope and y-intercept 5? Plan Which linear equation form should you use? B What is an equation of the line through ( 1, 5) with slope 2? When you know the slope and the y-intercept, use slopeintercept form. When you know the slope and a point on the line, use point-slope form. Problem 5 Using Two Points to Write an Equation What is an equation of the line at the right? Plan What is the first thing you need to know? It doesn't matter yet what form of linear equation you plan to use. You'll need the slope for both slope-intercept form and point-slope form. Think Write y Start by finding the slope m of the line through the given points. m= 2 y 1 5 ( 1) 6 x 2 x 1 = = ( 2) 5 You have the slope, and you know two points on the line. Use point-slope form. y y 1 = m(x x 1) 6 Use either point for (x 1, y 1). For example, you can use (, 5). y 5= (x ) 5 Problem 6 Writing Equations of Horizontal and Vertical Lines What are the equations for the horizontal and vertical lines through (2, 4)? Every point on the horizontal line through (2, 4) has a y-coordinate of 4. The equation of the line is y = 4. It crosses the y-axis at (0, 4). Every point on the vertical line through (2, 4) has an x-coordinate of 2. The equation of the line is x = 2. It crosses the x-axis at (2, 0). Think How is this different from writing other linear equations? You don't need the slope. Just locate the point where the line crosses the x-axis (for vertical) or y-axis (for horizontal).

Page 1 of 2 PRACTICE and APPLICATION EXERCISES Find the slope of the line passing through the given points. Scan page for a Virtual Nerd tutorial video. 1. For additional support when completing your homework, go to PearsonTEXAS.com. 2.. (, 7), ( 1, 4) 4. (, 2), ( 6, 2) 5. (5, 9), (5, 6) Graph each line. 6. y = x + 2 5 7. y = x +2 1 8. y = (x ) 9. y 1 = (x + 2) Use the given information to write an equation of each line. 2 10. slope, passes through ( 2, 6) 11. slope, passes through (4, 1) 12.

Page 2 of 2 1. 14. passes through (0, 5) and (5, 8) 15. passes through (6, 2) and (2, 4) 16. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Derive the slope formula. Include a graph as part of your answer. For Exercises 17 and 18, write the equation of the horizontal and vertical lines though the given point. 17. (4, 7) 18. (0, 1) 19. Write equations for three lines that contain the point (5, 6). Graph each pair of lines. Then find their point of intersection. 20. y = 4, x = 6 21. x = 0, y = 0 22. x = 1, y = PearsonTEXAS.com Page 127 Copyright 2016 Pearson Education, Inc. or its affiliate(s). All rights reserved. Privacy Policy Terms of Use Rights and Permissions

Page 1 of 2 2. What is the slope of the x-axis? Explain. Write an equation for the x-axis. 24. What is the slope of the y-axis? Explain. Write an equation for the y-axis. 25. Analyze Mathematical Relationships (1)(F) You want to construct a funbox at a local skate park. The skate park's safety regulations allow for the ramp on the funbox to have a maximum slope of. If you use the funbox plan at the right, can you build the ramp to meet the safety regulations? Explain. 4 11 Write each equation in slope-intercept form. 26. y 5 = 2(x + 2) 27. y + 2 = (x 4) 28. x + 2y = 10 1 STEM 29. Apply Mathematics (1)(A) The equation P = d +1represents the pressure P in atmospheres a scuba diver feels d feet below the surface of the water. a. What is the slope of the line? What does the slope represent in this situation? b. What is the y-intercept (P-intercept)? What does it represent in this situation? 0. Analyze Mathematical Relationships (1)(F) The x-intercept of a line is 2, and the y-intercept is 4. Use this information to write an equation for the line. 1. Connect Mathematical Ideas (1)(F) The vertices of a triangle are A(0, 0), B(2, 5), and C(4, 0). a. Write an equation for the line through A and B. b. Write an equation for the line through B and C. c. Compare the slopes and the y-intercepts of the two lines. Do the three points lie on one line? Justify your answer. 2. (5, 6), (, 2), (6, 8). ( 2, 2), (4, 4), (0, 0) 4. (5, 4), (2, ), ( 1, 10) TEXAS Test Practice 5. AB has endpoints A(8, k) and B(7, ). The slope of AB is 5. What is k? A. 1 B. 2 C. 5 D. 8 6. Two angles of a triangle measure 68 and 54. What is the measure of the third angle? F 14 G. 58 H. 122 J. 180 7. One of the angles in a certain linear pair is acute. Your friend says the other angle must be obtuse. Is your friend's conjecture reasonable? Explain. Page 128