UNIT 2.4 MORE ABOUT UNIT 2.4 MORE ABOUT LINEAR EQUATIONSLINEAR EQUATIONS

Warm Up

Write each function in slope-intercept form.

1. 4x + y = 8 2. –y = 3x 3. 2y = 10 – 6x

Determine whether each line is vertical or horizontal.

4. 5. y = 0

y = –4x + 8 y = –3x y = –3x + 5

3 4x =

vertical horizontal

Use slope-intercept form and point-slope form to write linear functions.Write linear functions to solve problems.

Objectives

Point-slope form

Vocabulary

Recall from Lesson 2-3 that the slope-intercept form of a linear equation is y= mx + b, where m is the slope of the line and b is its y-intercept.

In Lesson 2-3, you graphed lines when you were given the slope and y-intercept. In this lesson you will write linear functions when you are given graphs of lines or problems that can be modeled with a linear function.

Write the equation of the graphed line in slope-intercept form.

Step 1 Identify the y-intercept.The y-intercept b is 3.

Check It Out! Example 1

Step 2 Find the slope.

Choose any two convenient points on the line, such as (–4, 0) and (0, 3). Count from (–4, 0) to (0, 3) to find the rise and the run. The rise is 3 units and the run is 4 units

Check It Out! Example 1 Continued

3

Slope is = . riserun

34

3

4 4

3

Step 3 Write the equation in slope-intercept form.

Check It Out! Example 1 Continued

y = mx + b

34

y = x + 3 m = and b = 3. 34

The equation of the line is 34

y = x + 3.

Notice that for two points on a line, the rise is the differences in the y-coordinates, and the run is the differences in the x-coordinates. Using this information, we can define the slope of a line by using a formula.

If you reverse the order of the points in Example 2B, the slope is still the same.

m = =

Helpful Hint

6 – 165 – 11

– 10– 6

53

Find the slope of the line.x –6 –4 –2

y –3 –1 1

Let (x1, y1) be (–4, –1) and (x2, y2) be (–2, 1).

Choose any two points.

Use the slope formula.

The slope of the line is 1.

Check It Out! Example 2A

Let (x1, y1) be (2, –5) and (x2, y2) be (–3, –5).

The slope of the line is 0.

Check It Out! Example 2B

Find the slope of the line through (2,–5) and (–3, –5).

Use the slope formula.

Because the slope of line is constant, it is possible to use any point on a line and the slope of the line to write an equation of the line in point-slope form.

Method A Point-Slope Form

Rewrite in slope-intercept form.

Substitute.

Simplify.

Solve for y.

Distribute.

Write the equation of the line in slope-intercept form with slope –5 through (1, 3).

Check It Out! Example 3a

The equation of the slope is y = –5x + 8.

y – y1 = m(x – x1)

y – (3) = –5(x – 1)

y – 3 = –5(x – 1)

y – 3 = –5(x – 1)

y – 3 = –5x + 5

y = –5x + 8

Method B Slope-Intercept Form

Check It Out! Example 3bWrite the equation of the line in slope-intercept form through (–2, –3) and (2, 5).

First, find the slope. Let (x1, y1) be (–2,–3) and (x2, y2) be (2, 5).

y = mx + b

5 = (2)2 + b

5 = 4 + b

1 = b

Rewrite the equation using m and b.

y = mx + b y = 2x + 1

The equation of the line is y = 2x + 1.

Items Cost ($)

4 14.00

7 21.50

18

Express the cost as a linear function of the number of items.

Let x = items and y = cost.

Find the slope by choosing two points. Let (x1, y1) be (4, 14) and (x2, y2) be (7, 21.50).

Check It Out! Example 4a

To find the equation for the number of items, use point-slope form.

Use the data in the first row of the table.

Simplify.

Check It Out! Example 4a Continued

y – y1 = m(x – x1)

y – 14 = 2.5(x – 4)

y = 2.5x + 4

Graph the relationship between the number of items and the cost. Find the cost of 18 items.

To find the cost, use the graph or substitute the number of items into the function.

Substitute.

The cost for 18 items is $49.

Check It Out! Example 4b

y = 2.5(18) + 4

y = 45 + 4

y = 49

By comparing slopes, you can determine if the lines are parallel or perpendicular. You can also write equations of lines that meet certain criteria.

Write the equation of the line in slope-intercept form.

parallel to y = 5x – 3 and through (1, 4)

Parallel lines have equal slopes.

Use y – y1 = m(x – x1) with (x1, y1) = (5, 2).

Distributive property.

Simplify.

m = 5

y – 4 = 5(x – 1)

y – 4 = 5x – 5

y = 5x – 1

Check It Out! Example 5a

Distributive property.

Simplify.

Use y – y1 = m(x – x1). y + 2 is equivalent to y – (–2).

Check It Out! Example 5b

The slope of the given line is , so the slope of

the perpendicular, line is the opposite reciprocal .

Write the equation of the line in slope-intercept form.

perpendicular to and through (0, –2)

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