MR. JONES - Lesson 5-2 Writin uations in Standard and ......Step 1 Plot the given point (-2, 7). Step 2 Use the slope, - __ 3 2, to plot another point on the line. Step 3 Draw a line

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Talk About It!

Why must a line be nonvertical in order to be written in point-slope form? Explain.

Writing Equations in Standard and Point-Slope Forms

Lesson 5-2

Today’s Vocabulary point-slope formparallel linesperpendicular lines

Explore Forms of Linear Equations

Online Activity Use graphing technology to complete the Explore.

Learn Creating Linear Equations in Point-Slope FormWhen the slope and the coordinates of one point of a line are known, an equation for the line can be written in point-slope form.

Key Concept • Point-Slope Form

Words The linear equation y - y1 = m(x - x1) is written in point-slope form, where (x1, y1) is a given point on a nonvertical line and m is the slope of the line.

Symbols y - y1 = m(x - x1)Example y

xO

(x1, y1)

If you are given two points on the line or a point on the line and its slope, you can write an equation for the line in point-slope form.

Key Concept • Writing Equations of Lines in Point-Slope FormGiven the Slope and One Point Given Two Points

Step 1 Let the x and y coordinates be (x1, y1).

Step 1 Find the slope.

Step 2 Substitute the values of m, x1, and y1 into the equation of a line in point-slope form.

Step 2 Chose one of the two points to use.

Step 3 Follow the steps for writing an equation given the slope and one point.

INQUIRY How are the point-slope and slope-intercept forms of a linear equation related?

Today’s Goals● Write equations of lines

in point-slope form.● Create and identify

equations of parallel or perpendicular lines.

Lesson 5-2 • Writing Equations in Standard and Point-Slope Forms 295

Sample answer: Vertical lines have undefined slopes. Because slope is a key element of a point-slope equation, the line must have a real number slope in order to be written in point-slope form.

THIS MATERIAL IS PROVIDED FOR INDIVIDUAL EDUCATIONAL PURPOSES ONLY AND MAY NOT BE DOWNLOADED, REPRODUCED, OR FURTHER DISTRIBUTED.

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Example 1 Equation in Point-Slope Form Given Slope and a PointWrite an equation in point-slope form for the line that passes through (-2, 7) with a slope of - 3 __ 2 . Then graph the equation.

y - y1 = m(x - x1) Point-slope form

y - 7 = - 3 __ 2 [x - (-2)] (x1, y1) = ( , ) and m =

Simplify.

Step 1 Plot the given point (-2, 7).

Step 2 Use the slope, - 3 __ 2 , to plot another point on the line.

Step 3 Draw a line through the points.y

O x

Check Determine the equation in point-slope form for the line that passes through (7, 5) with a slope of -3. Then graph the equation.

y = (x )

O

y

x−8 −4−2−6 84 62

8

46

2

−8

−4−6

Go Online You can complete an Extra Example online.

Your Notes

296 Module 5 • Creating Linear Equations

- 3 __ 2

y - 7 = - 3 __ 2 (x + 2)

-2 7

- 5 -3 - 7


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Think About It!

Write another equation in point-slope form for the line with the points given.

Why are there multiple correct answers with the same given information?

Study Tip

Checking Your WorkTo check your work, you can substitute the point from the original point-slope form of the equation, in this case (6, -4), into the slope-intercept form of the equation. If it is a true statement, the equation is correct.

y = -2x + 8-4 = -2(6) + 8-4 = -12 + 8-4 = -4 ✓

Example 2 Equation in Point-Slope Form Given Two PointsWrite an equation in point-slope form for the line that passes through the given points.

(2, -7) and (6, -3)


m = y2 - y1 _____ x2 - x1

Slope Formula

m = -3 - (-7)

_______ 6 - 2 = 4 _ 4 or (x1, y1) = (2, -7) and (x2, y2) = (6, -3)

Step 2 Write an equation.

You can select either point for (x1, y1) in point-slope form.

y - y1 = m(x – x1) Point-slope form

y - = 1(x - ) (x1, y1) = (6, -3) and m = 1

Simplify.

Check Select an equation in point-slope form for the line that passes through (-16, 18) and (-11, -2).

A. y + 2 = -4(x + 11)

B. y + 2 = - 1 _ 4 (x + 11)

C. y - 2 = -4(x + 11)

D. y - 2 = - 1 _ 4 (x - 11)

E. None of these

Example 3 Change to Slope-Intercept FormWrite y + 4 = -2(x - 6) in slope-intercept form.

y + 4 = -2(x - 6) Original Equation

y + 4 = -2x + 12

y = -2x + 8 Subtract from each side.

Check Write y + 3 = - 1 __ 2 (x – 8) in slope-intercept form.

y = x +



1

A

y + 3 = (x - 6)

(-3) 6

Distributive Property

4

1- 1 __ 2

y + 7 = (x ‒ 2)

Sample answer: Only a slope and one point on a line are required to write an equation of the line. Any point on a line can be used to write the equation. So, if two points are given, either point can be used to write an equivalent equation.


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Think About It!

Use the equation to find the cost of purchasing 12 books.

Study Tip

Fractional SlopesWhen working with an equation with a fractional slope, it is often simpler to first multiply each side of the equation by the denominator. This will eliminate distributing a fraction later in the equation.

Example 4 Apply Point-Slope FormREADING Nadia’s book club is ordering new novels. She knows that the total cost of 5 books is $61.25, and 15 books cost $159.75. Write an equation in point-slope form to represent the total cost of the books ordered.


m = y2 - y1 ______ x2 - x1

Slope Formula

m = 159.75 - 61.25 __________ 15 - 5 = 98.5 ____ 10 or 9.85 (x1, y1) = (5, 61.25) and (x2, y2) = (15, 159.75)

Step 2 Write an equation.

y - y1 = m(x - x1) Point-slope form

y - = 9.85(x - ) (x1, y1) = (5, 61.25) and m = 9.85

Simplify.

Check TAXIS The total cost of a taxi fare is given in the table. Determine the equation(s) in point-slope form that model(s) this situation if x represents the distance in miles and y represents the cost in dollars.

Distance (miles) 1.5 4 7.5 12.25

Cost (dollars) 6.90 13.40 22.50 34.85

A. y - 13.4 = 2.6(x - 4) B. y - 22.5 = 2.6(x ‒ 7.5)

C. y = 2.6x + 3 D. y - 6.9 = 5 __ 13 (x - 1.5)

E. y + 34.85 = 2.6(x + 12.25)

Example 5 Change to Standard FormWrite y - 1 = - 2 __ 5 (x + 3) in standard form.

y - 1 = - 2 __ 5 (x + 3) Original equation

(y - 1) = (x + 3) Multiply each side by 5 to eliminate the fraction.

Distributive Property

5y = -2x - 1 Add 5 to each side.

Add 2x to each side.

Check Write y = - 7 __ 2 x + 5 in standard form.



$130.20

y - 61.25 = 9.85 (x - 5)

61.25 5

A, B

-25

5y - 5 = -2x - 6

2x + 5y = -1

7x + 2y = 10


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Go OnlineAn alternate method is available for this example.

Example 6 Standard Form Given Two PointsWrite an equation in standard form for the line that passes through (8, -4) and (-6, -11).


m = y2 - y1 ______ x2 - x1

Slope Formula

m = -11 - (-4)

________ -6 - 8 = -7 ___ -14 or (x1, y1) = (8, -4) and (x2, y2) = (-6, -11)

Step 2 Write an equation in slope-intercept form.

y = mx + b Slope-intercept form

-4 = 1 __ 2 (8) + b (x, y) = ( , ) and m =

-4 = + b Simplify.

-8 = b Subtract 4 from each side.

y = Replace m with 1 __ 2 and b with -8.

Step 3 Write the equation in standard form.

2y = 2 ( 1 __ 2 x - 8) Multiply each side by 2.

2y = Distributive Property

-x + 2y = -16 Subtract x from each side.

Multiply each side by -1.

Check Select the equation in standard form for the line that passes through (-9, 8) and (1, -12).

A. x + 2y = -20

B. 2x + y = -13

C. 2x + y = 7

D. 2x + y = -10

E. x + 2y = 20



1 __ 2

1 __ 2 x - 8

4

8

x - 2y = 16

x - 16

-4 1 __ 2

D


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Think About It!

If the given line is vertical, what is the slope of any line parallel to the given line? perpendicular to the given line?

Learn Equations of Parallel and Perpendicular Lines Nonvertical lines in the same plane that have the same slope are called parallel lines. Nonvertical lines in the same plane for which the product of the slopes is -1 are called perpendicular lines.

Key Concept • Slopes of Parallel and Perpendicular LinesParallel Lines Perpendicular Lines

If two nonvertical lines are parallel, their slopes are the same.

y

O x

C

A

B

D

Since both lines have a slope of 1 __ 2 , ⟷ AB ∥

⟷ CD .

If two nonvertical lines are perpendicular, the product of their slopes is -1.

y

O

BE

Ax

F

Since 1 __ 2 (-2) = -1, ⟷ AB ⊥ ⟷ EF .

You can write an equation of a line parallel or perpendicular to a given line if you know a point on the line and an equation of the given line.

Key Concept • Writing Equations of Lines Parallel or Perpendicular to a Given LineParallel Lines Perpendicular Lines

Step 1 Identify the slope m of the given line.

Step 1 Identify the slope m of the given line. The slope of the line perpendicular to the original line is - 1 __ m .

Step 2 Use the point-slope form with slope m and the coordinates of the given point.

Step 2 Use the point-slope form with slope - 1 __ m and the coordinates of the given point.

Step 3 Rewrite the equation in the needed form.

Step 3 Rewrite the equation in the needed form.

Example 7 Parallel Line Through a Given PointWrite an equation in slope-intercept form for the line that passes through (-4, 2) and is parallel to the graph of y = 3x - 5.

Step 1 Identify the slope of the given line.

The slope of the line with equation y = 3x - 5 is . The line parallel to that line has the same slope, .


Go OnlineYou may want to complete the Concept Check to check your understanding.


Sample answer: undefined slope; 0

33


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Study Tip

Checking Your WorkTo check that your equation represents the correct line, graph both lines. Verify that the lines appear to be parallel and that your line passes through the given point.

Steps 2, 3 Write the equation of the parallel line.

Use the point-slope form to rewrite the equation in slope-intercept form. y - y1 = m(x – x1) Point-slope form

y - = [x - (-4)] (x1, y1) = (-4, 2) and m = 3

y - = Simplify.

y - 2 = Distributive Property

y = 3x Add 2 to each side.

Check Write an equation for the line that passes through (8, 2) and is parallel to the graph of y = 3 _ 4 x + 2.

Example 8 Perpendicular Line Through a Given PointWrite an equation in slope-intercept form for the line that passes through (1, -2) and is perpendicular to the graph of 3x + 2y = 12.

Step 1 Identify the slope of the given line.

Write the equation in slope-intercept form. 3x + 2y = 12 Original equation

3x + 2y = 12 Subtract 3x from each side.

2y = + 12 Simplify.

2y

__ 2 = -3x ___ 2 + 12 __ 2 Divide each side by 2.

y = Simplify.

The slope of the line with equation 3x + 2y = 12 is - 3 __ 2 . The slope of the line perpendicular to that line is the opposite

reciprocal, .

Steps 2, 3 Write the equation of the perpendicular line.Use the point-slope form to rewrite the equation in slope-intercept form.

y - y1 = m(x – x1) Point-slope form

y - = (x - 1) (x1, y1) = (1, -2) and m = 2 __ 3

y + = 2 __ 3 (x ) Simplify.

y + 2 = Distributive Property

y = 2 __ 3 x Subtract 2 from each side.



3

3(x + 4)

3x + 12

y = 3 __ 4 x - 4

2 __ 3

-3x

- 3 __ 2 x + 6

(-2)

- 1 2 __ 3 x - 2 __ 3

- 8 __ 3

2 __ 3

2

- 3x - 3x

+ 14

2

2


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Check Select the equation in slope-intercept form for the line that passes through (5, 0) and is perpendicular to the graph of x - 6y = 1. A. y = -6x + 30B. 6x + y = 30C. y = - 1 __ 6 x + 2D. x - 6y = 5

Example 9 Determine Line RelationshipsDetermine whether ⟷ AB and ⟷ EF are parallel, perpendicular, or neither for A(6, 8), B(2, 5), E(-6, -3), and F(0, 5).

Step 1 Find the slope of each line.

slope of ⟷ AB = 8 - 5 ____ 6 - 2 = slope of ⟷ EF = -3 - 5 ______ -6 - 0 = -8 ___ -6 or

Step 2 Determine the relationship.

parallel To determine whether the lines are parallel, compare their slopes. The two lines have the same slope, so they parallel.

perpendicular To determine whether the lines are perpendicular, find the product of their slopes. 3 _ 4 · 4 _ 3 =

Since the product of the slopes -1, ⟷ AB and ⟷ EF perpendicular.

Check Determine whether

⟷ CD and ⟷ KL are parallel, perpendicular, or neither

for C(4, 10), D(-1, 12), K(6, -5), and L(1, -3).

⟷

CD and ⟷ KL are .

Complete each sentence given y = ax - 5 and y = bx + 3.

When a = 4 and b = 4, the graphs are .

When a = -3 and b = 5, the graphs are .

When a = -2 and b = 1 __ 2 , the graphs are .


Go Online to practice what you’ve learned about writing linear equations in the Put It All Together over Lessons 5-1 and 5-2.


3 __ 4

do notare not

are not

1

4 __ 3

A

is not

parallel

parallelneither

perpendicular


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MR. JONES - Lesson 5-2 Writin uations in Standard and ......Step 1 Plot the given point (-2, 7). Step 2 Use the slope, - __ 3 2, to plot another point on the line. Step 3 Draw a line