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# 5.5 parallel and perpendicular lines (equations) day 1

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### 5.5 parallel and perpendicular lines (equations) day 1

1. 1. Parallel and Perpendicular Lines (Equations) Section 5.5 P. 319
2. 2. Key Concept (and its converse)If two nonvertical lines in the same plane have the same slope, then they are parallel. (from Chpt.4)If two nonvertical lines in the same plane are parallel, then they have the same slope.
3. 3. EXAMPLE 1 Write an equation of a parallel line Write an equation of the line that passes through (–3,–5) and is parallel to the line y = 3x – 1.SOLUTION STEP 1 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.STEP 2Find the y-intercept. Use the slope and the given point.
4. 4. EXAMPLE 1 Write an equation of a parallel line y = mx + b Write slope-intercept form. – 5 = 3(– 3) + b Substitute 3 for m, 23 for x, and 25 for y. 4=b Solve for b. STEP 3 Write an equation. Use y = mx + b. y = 3x + 4 Substitute 3 for m and 4 for b.
5. 5. GUIDED PRACTICE for Example 11. Write an equation of the line that passes through (–2, 11) and is parallel to the line y = – x + 5.SOLUTIONSTEP 1Identify the slope. The graph of the given equationhas a slope of – 1.So, the parallel line through (– 2, 11)has a slope of – 1.STEP 2Find the y-intercept. Use the slope and the given point.
6. 6. GUIDED PRACTICE for Example 1 y = mx + b Write slope-intercept form. 11 = (–1 )(– 2) + b Substitute 11 for y, – 1 for m, and – 2 for x. 9=b Solve for b.STEP 3Write an equation. Use y = m x + b. y =–x+9 Substitute – 1 for m and 9 for b.
7. 7. Key ConceptIf two nonvertical lines in the same plane have slopes that are negative reciprocals, then the lines are perpendicular.If two nonvertical lines in the same plane are perpendicular, then their slope are negative reciprocals.
8. 8. • Here is an example of two lines that ARE perpendicular: y = - 4x + 6 y=¼x -3 Note: ¼ and -4 are negative reciprocals
9. 9. EXAMPLE 4 Write an equation of a perpendicular line Write an equation of the line that passes through (4, – 5) and is perpendicular to the line y = 2x + 3. SOLUTION STEP 1 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
10. 10. EXAMPLE 4 Write an equation of a perpendicular lineSTEP 2 Find the y-intercept. Use the slope and the given point. y = mx + b Write slope-intercept form. 1 – 5 = – 1 (4) + b Substitute – for m, 4 for x, and 2 2 – 5 for y. –3= b Solve for b.STEP 3 Write an equation. y=mx+b Write slope-intercept form. 1 1 y= – 2x – 3 Substitute – for m and – 3 for b. 2
11. 11. GUIDED PRACTICE for Examples 3 and 43. Is line “a” perpendicular to line “b”? Justify youranswer using slopes Line a: 2y + x = – 12 Line b: 2y = 3x – 8 SOLUTIONFind the slopes of the lines. Write the equations inslope-intercept form.Line a: 2y + x = 12 Line b: 2y = 3x -8 1 y = 3/2x -4 y=– x–6 2
12. 12. EXAMPLE 2 Determine whether lines are parallel or perpendicular Determine which lines, if any, are parallel or perpendicular. Line a: y = 5x – 3 Line b: x +5y = 2 Line c: –10y – 2x = 0 SOLUTION Find the slopes of the lines. Line a: The equation is in slope-intercept form. The slope is 5. Write the equations for lines b and c in slope- intercept form.
13. 13. EXAMPLE 2 Determine whether lines are parallel or perpendicular Line b: x + 5y = 2 5y = – x + 2 –1 x 2 y= 5 + 5 Line c: – 10y – 2x = 0 – 10y = 2x 1 y= – 5x
14. 14. EXAMPLE 2 Determine whether lines are parallel or perpendicular ANSWER Lines b and c have slopes of – 1 , so they are 5 parallel. Line a has a slope of 5, the negative reciprocal of – 1 , so it is perpendicular to lines b and c. 5
15. 15. GUIDED PRACTICE for Example 2Determine which lines, if any, are parallel orperpendicular.Line a: 2x + 6y = – 3Line b: 3x – 8 = yLine c: –1.5y + 4.5x = 6Find the slopes of the lines. Line a: 2x + 6y = – 3 6y = –2x – 3 y= – 1x – 1 3 2
16. 16. GUIDED PRACTICE for Example 2Line b: 3x – 8 = yLine c: –1.5y + 4.5x = 6 – 1.5y = 4.5x – 6 y = 3x – 4 Lines b and c have slopes of 3, so they are parallel. Line a has a slope of – 1 , the negative reciprocal 3 of 3, so it is perpendicular to lines b and c.
17. 17. Assignment:• P. 322 1, 2, 6-8, 12-14, 18-20, 32
18. 18. • Things to study for the test• Sections 5.1 – 5.5 (omit 5.3)• Write an equation in slope-intercept form given the slope and y –int• Write an equation in slope-intercept form given the graph or two points• Standard Form – given two points• Parallel Lines – write equations• Perpendicular Lines – write equations