Equations of Lines

LESSON 3–4

Five-Minute Check (over Lesson 3–3)

TEKS

Then/Now

New Vocabulary

Key Concept: Nonvertical Line Equations

Example 1: Slope and y-intercept

Example 2: Slope and a Point on the Line

Example 3: Two Points

Example 4: Horizontal Line

Key Concept: Horizontal and Vertical Line Equations

Example 5: Write Equations of Parallel or Perpendicular Lines

Example 6: Real-World Example: Write Linear Equations

Over Lesson 3–3

A.

B.

C.

D.

What is the slope of the line MN for M(–3, 4) and N(5, –8)?

Over Lesson 3–3

A.

B.

C.

D.

What is the slope of a line perpendicular to MN for M(–3, 4) and N(5, –8)?

Over Lesson 3–3

A.

B.

C.

D.

What is the slope of a line parallel to MN forM(–3, 4) and N(5, –8)?

Over Lesson 3–3

A. B.

C. D.

What is the graph of the line that has slope 4 and contains the point (1, 2)?

Over Lesson 3–3

What is the graph of the line that has slope 0 and contains the point (–3, –4)?

A. B.

C. D.

Over Lesson 3–3

A. (–2, 2)

B. (–1, 3)

C. (3, 3)

D. (4, 2)

Targeted TEKSG.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.G.2(C) Determine an equation of a line parallel or perpendicular to a given line that passes through a given point.

Mathematical ProcessesG.1(B), G.1(D)

You found the slopes of lines.

• Write an equation of a line given information about the graph.

• Solve problems by writing equations.

• slope-intercept form

• point-slope form

Slope and y-intercept

Write an equation in slope-intercept form of the line with slope of 6 and y-intercept of –3. Then graph the line.

y = mx + b Slope-intercept form

y = 6x + (–3) m = 6, b = –3

y = 6x – 3 Simplify.

Slope and y-intercept

Answer: Plot a point at the y-intercept, –3.

Use the slope of 6 or to find

another point 6 units up and1 unit right of the y-intercept.

Draw a line through these two points.

A. x + y = 4

B. y = x – 4

C. y = –x – 4

D. y = –x + 4

Write an equation in slope-intercept form of the line with slope of –1 and y-intercept of 4.

Slope and a Point on the Line

Point-slope form

Write an equation in point-slope form of the line

whose slope is that contains (–10, 8). Then

graph the line.

Simplify.

Slope and a Point on the Line

Answer: Graph the given point (–10, 8).

Use the slope

to find another point 3 units down and 5 units to the right.

Draw a line through these two points.

Write an equation in point-slope form of the line

whose slope is that contains (6, –3).

A.

B.

C.

D.

Two Points

A. Write an equation in slope-intercept form for a line containing (4, 9) and (–2, 0).

Step 1 First, find the slope of the line.

Slope formula

x1 = 4, x2 = –2, y1 = 9, y2 = 0

Simplify.

Two Points

Step 2 Now use the point-slope form and either point to write an equation.

Distributive Property

Add 9 to each side.

Answer:

Point-slope form

Using (4, 9):

Two Points

B. Write an equation in slope-intercept form for a line containing (–3, –7) and (–1, 3).

Step 1 First, find the slope of the line.

Slope formula

x1 = –3, x2 = –1, y1 = –7, y2 = 3

Simplify.

Two Points

Step 2 Now use the point-slope form and either point to write an equation.

Distributive Property

Answer:

m = 5, (x1, y1) = (–1, 3)

Point-slope form

Using (–1, 3):

Add 3 to each side.y = 5x + 8

A. Write an equation in slope-intercept form for a line containing (3, 2) and (6, 8).

A.

B.

C.

D.

A. y = 2x – 3

B. y = 2x + 1

C. y = 3x – 2

D. y = 3x + 1

B. Write an equation in slope-intercept form for a line containing (1, 1) and (4, 10).

Horizontal Line

Write an equation of the line through (5, –2) and (0, –2) in slope-intercept form.

Slope formula

This is a horizontal line.

Step 1

Horizontal Line

Point-Slope form

m = 0, (x1, y1) = (5, –2)

Step 2

Answer:

Simplify.

Subtract 2 from each side.y = –2

Write an equation of the line through (–3, 6) and (9, –2) in slope-intercept form.

A.

B.

C.

D.

Write Equations of Parallel or Perpendicular Lines

y = mx + b Slope-Intercept form

0 = –5(2) + b m = –5, (x, y) = (2, 0)

0 = –10 + b Simplify.

10 = b Add 10 to each side.

Answer: So, the equation is y = –5x + 10.

A. y = 3x

B. y = 3x + 8

C. y = –3x + 8

D.

Write Linear Equations

RENTAL COSTS An apartment complex charges $525 per month plus a $750 annual maintenance fee. A. Write an equation to represent the total first year’s cost A for r months of rent.

For each month of rent, the cost increases by $525. So the rate of change, or slope, is 525. The y-intercept is located where 0 months are rented, or $750.

A = mr + b Slope-intercept form

A = 525r + 750 m = 525, b = 750

Answer: The total annual cost can be represented by the equation A = 525r + 750.

Write Linear Equations

RENTAL COSTS An apartment complex charges $525 per month plus a $750 annual maintenance fee.

Evaluate each equation for r = 12.

First complex: Second complex:A = 525r + 750 A = 600r + 200

= 525(12) + 750 r = 12 = 600(12) + 200= 7050 Simplify. = 7400

B. Compare this rental cost to a complex which charges a $200 annual maintenance fee but $600 per month for rent. If a person expects to stay in an apartment for one year, which complex offers the better rate?

Write Linear Equations

Answer: The first complex offers the better rate: one year costs $7050 instead of $7400.

A. C = 25 + d + 100

B. C = 125d

C. C = 100d + 25

D. C = 25d + 100

RENTAL COSTS A car rental company charges $25 per day plus a $100 deposit.

A. Write an equation to represent the total cost C for d days of use.

A. first company

B. second company

C. neither

D. cannot be determined

RENTAL COSTS A car rental company charges $25 per day plus a $100 deposit.

B. Compare this rental cost to a company which charges a $50 deposit but $35 per day for use. If a person expects to rent a car for 9 days, which company offers the better rate?

Equations of Lines

LESSON 3–4