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Warm Up

Learn how to write equations of parallel and perpendicular lines with step-by-step examples and practice problems in this interactive lesson. Determine if lines are parallel or perpendicular using slopes.

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Warm Up

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  1. Write Equations of Parallel and Perpendicular Lines Warm Up Lesson Presentation Lesson Quiz

  2. Warm-Up Are the lines parallel? Explain. 1. y– 2 = 2x, 2x + y = 7 No; one slope is 2 and the other is–2. ANSWER 2. –x = y + 4, 3x + 3y = 5 ANSWER Yes; both slopes are–1.

  3. Warm-Up 3. You play tennis at two clubs. The total cost C(in dollars) to play for time t(in hours) and rent equipment is given by C= 15t + 23 at one club and C= 15t + 17 at the other. What is the difference in total cost after 4 hours of play? ANSWER $6

  4. Example 1 Write an equation of the line that passes through (–3, –5) and is parallel to the liney = 3x – 1. SOLUTION STEP1 Identify the slope. The graph of the given equation has a slope of 3. So, the parallel line through (–3, –5) has a slope of 3.

  5. Example 1 STEP2 Find the y-intercept. Use the slope and the given point. y=mx+b Write slope-intercept form. –5=3(–3) +b Substitute3form,3for x,and5fory. Solve forb. 4 = b STEP3 Write an equation. Usey = mx + b. Substitute3formand4forb. y = 3x + 4

  6. 1. Write an equation of the line that passes through (–2, 11) and is parallel to the liney =–x + 5. ANSWER y = –x + 9 Guided Practice

  7. SOLUTION Find the slopes of the lines. Line a: The equation is in slope-intercept form.The slope is 5. Example 2 Determine which lines, if any, are parallel or perpendicular. Line a: y = 5x – 3 Line b:x + 5y = 2 Line c:–10y – 2x = 0 Write the equations for lines band cin slope-intercept form.

  8. Lineb: ANSWER x + 5y = 2 Lines band chave slopes of – , so they are parallel. Lineahas a slope of5,the negative reciprocal of – , so it is perpendicular to lines band c. – x y = + 2 1 1 1 1 Linec: –10y – 2x = 0 5 5 5 5 5 x – y = Example 2 –10y = 2x 5y = – x + 2

  9. ANSWER parallel: b and c; perpendicular: a and b, a and c Guided Practice Determine which lines, if any, are parallel or perpendicular. Line a: 2x + 6y =–3 Line b: y = 3x – 8 Line c:–1.5y + 4.5x = 6

  10. STATE FLAG The Arizona state flag is shown in a coordinate plane. Lines aand bappear to be perpendicular. Are they? Example 3 Linea:12y = –7x + 42 Lineb:11y = 16x – 52 SOLUTION Find the slopes of the lines. Write the equations in slope-intercept form.

  11. 7 16 7 16 11 12 11 12 x y=– + 42 52 12 11 x y = – ANSWER The slope of line ais –. The slope of line bis . The two slopes are not negative reciprocals, so lines aand bare not perpendicular. Example 3 Linea:12y = –7x + 42 Lineb:11y = 16x – 52

  12. ANSWER 1 No; the slope of line ais –, the slope of line bis . The slopes are not negative reciprocals so the lines are not perpendicular. 2 3 2 Guided Practice 3. Is line a perpendicular to line b?Justify your answer using slopes. Linea:2y +x = –12 Lineb:2y = 3x – 8

  13. Identify the slope. The graph of the given equation has a slope of 2. Because the slopes of perpendicular lines are negative reciprocals, the slope of the perpendicular line through (4, –5) is . 1 2 – Example 4 Write an equation of the line that passes through(4, –5)and is perpendicular to the liney = 2x + 3. SOLUTION STEP1

  14. y= mx+b 1 – (4) +b –5= Substitute – for m, 4 for x, and –5 for y. 2 1 2 –3= b 1 2 1 y = – x – 3 Substitute – formand–3for b. 2 Example 4 STEP 2 Find the y-intercept. Use the slope and the given point. Write slope-intercept form. Solve for b. STEP3 Write an equation. y = mx + b Write slope-intercept form.

  15. 1 4 y = – x + 4 ANSWER Guided Practice 4. Write an equation of the line that passes through (4, 3) and is perpendicular to the line y = 4x – 7.

  16. 1. Write an equation of the line that passes through the point (–1, 4) and is parallel to the line y = 5x – 2. ANSWER y = 5x + 9 2. Write an equation of the line that passes through the point (–1, –1) and is perpendicular to the line y = x + 2. 1 ANSWER – 4 y = 4x + 3 Lesson Quiz

  17. 3. Path a, b and c are shown in the coordinate grid. Determine which paths, if any, are parallel or perpendicular. Justify your answer using slopes. ANSWER 1 – 2 Paths a andb are perpendicular because their slopes, 2 andare negative reciprocals. No paths are parallel. Lesson Quiz

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