Warm-Up1) Find the slope of the line containing the points (-1,12) and (5,-6).

4 minutes

2) Identify the slope and y-intercept for the line y = -5x + 7.

3) Write the equation in slope-intercept form for the line that has a y-intercept of 0 and a slope of -1.

1.3.1 Linear Equations in Two 1.3.1 Linear Equations in Two VariablesVariables

1.3.1 Linear Equations in Two 1.3.1 Linear Equations in Two VariablesVariables

Objectives: •Write a linear equation in two variables given sufficient information

Example 1Write an equation for the line containing the points (1,5) and (2,8).

change in y-coordinatesm

change in x-coordinates

2 1

2 1

y ym

x x

8 5m

2 1

3m 3

1

Example 1Write an equation for the line containing the points (1,5) and (2,8).

m 3

y = mx + b5 =3(1)+

b5 = 3 + b-3 -32 = b

y = 3x + 2

substitute

simplify

The Point-Slope Equation

y – y1 = m(x – x1)

Example 2Write an equation for the line with slope 7 that contains the point (3,4).

y – y1 = m(x – x1)y – 4 = 7(x – 3)

y – 4 = 7x - 21+4

+4y = 7x - 17

Example 3Write an equation for the line containing (5,7) and (2,1).

y – y1 = m(x – x1)

First, find the slope: 2 1

2 1

y ym

x x

1 7m

2 5

63

2

y – 7 = 2(x – 5)

y – 7 = 2x - 10+7

+7y = 2x - 3

Example 4Write an equation for the line shown below.

-4 -2

2

42

4

-4

-2

First, find any two points on the line.

(-3,-3) and

(1,-1)

2m

4

12

y – y1 = m(x – x1) y + 3 = ½(x +

3)y + 3 = ½x + 1½ -3 -3

y = ½x – 1½

Example 5Marva left her house and drove at a constant speed to a conference in another state. She picked up Delia along the way. Two hours after picking up Delia, they were 140 miles from Marva’s house, and 5 hours after picking up Delia, they were 344 miles from Marva’s house. How far from her house was Marva when she picked up Delia?

Homework

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Warm-UpWrite an equation for the line given the indicated points.

4 minutes

1) (-6,-6) and (-3,1)

Write an equation in point-slope form for the line that has the indicated slope, m, and contains the given point.

2) m = 4; (9,-3)

1.3.2 Linear Equations in Two 1.3.2 Linear Equations in Two VariablesVariables

1.3.2 Linear Equations in Two 1.3.2 Linear Equations in Two VariablesVariables

Objectives: •Write an equation for a line that is parallel or perpendicular to a given line

ActivityGraph y = 2x + 1.

On the same screen, graph y = 2x, y = 2x – 2.5, and y = 3x + 1.

Which equations have graphs that appear to be parallel to that of y = 2x + 1?y = 2x, y = 2x –

2.5What do these equations have in common?same slopes

Write an equation in slope-intercept form for a line whose graph you think will be parallel to that of y = 2x + 1. Verify by graphing.

ActivityGraph y = 2x + 1.

On the same screen, graph

1y x 2,

2

1y x 2,

2 1

and y x 32

Which equations have graphs that appear to be perpendicular to that of y = 2x + 1?

1y x 2

2 1

and y x 32

What do these equations have in common?

their slopes are negative reciprocals of each other

Activity

Write an equation in slope-intercept form whose graph you think will be perpendicular to that of y = 2x + 1. Verify by graphing.

Parallel LinesParallel lines are lines in the same plane that never intersect.

-8 -6 -4 -2

2

42 6 8

4

6

-4

-6

-8

-2

8

Parallel lines have the same slope.

Perpendicular LinesPerpendicular lines are lines that intersect to form a 900 angle.

-8 -6 -4 -2

2

42 6 8

4

6

-4

-6

-8

-2

8

The product of the slopes of perpendicular lines is -1.

4m 2

2

2 1m

4 2

Example 1Determine whether these lines are parallel or perpendicular.

y – 2 = 5x + 4

and -15x + 3y = 9+2 +2

y = 5x + 6

+15x +15x 3y = 9 +

15x3 3

y = 3 + 5x

y = 5x + 3

The lines have the same slope.So they are parallel.

Example 2Write an equation in slope-intercept form for the line containing (-3,-5) and parallel to the line y = 2x + 1.First, we need the slope of the line y = 2x + 1.m = 2

Second, we need to find out the slope of the line that is parallel to y = 2x + 1.m 2

Lastly, we use the point-slope formula to find our equation.

1 1(y y ) m(x x ) (y 5) 2(x 3)

y 5 2(x 3) y + 5 = 2x + 6 y = 2x + 1

Example 3Write an equation for the line containing (-3,-5) and perpendicular to the line y = 2x + 1.First, we need the slope of the line y = 2x + 1.

m = 2Second, we need to find out the slope of the line that is perpendicular to y = 2x + 1. 1

m2

Lastly, we use the point-slope formula to find our equation.

1 1(y y ) m(x x ) 1

(y 5) (x 3)2

1y 5 (x 3)

2

1 3y 5 x

2 2

1 13y x

2 2

Homework

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