5.4 Write Linear Equations in Standard Form

•Students will write equations in standard form.

•Students will do assigned homework.

•Students will study

vocabulary words.

Lesson 5.4, For use with pages 311-316

ANSWER

ANSWER

1. (1, 4), (6, –1)

y + 2 = 3(x + 1) or y – 7 = 3(x – 2)

y – 4 = –(x – 1) or y + 1 = –(x – 6)

2. (–1, –2), (2, 7)

Write an equation in point-slope form of the line that passes through the given points.

3. A store rents 3 DVDs for $5, plus $3 for each additional DVD. Find the cost of renting 20 DVDs.

Lesson 5.4, For use with pages 311-316

ANSWER $56

ANSWER y + 4 = –2(x – 6)

Daily Homework Quiz

For use after Lesson 5.3

Write an equation in point-slope form of the line that passes through (6, –4) and has slope 2.

1.

Write an equation in point-slope form of the line that passes through (–1, –6) and (3, 10).

2.

ANSWER y + 6 = 4(x + 1) or y –10 = 4(x–3)

Daily Homework Quiz

For use after Lesson 5.3

A travel company offers guided rafting trips for $875 for the first three days and $235 for each additional day. Write an equation that gives the total cost (in dollars) of a rafting trip as a function of the length of the trip. Find the cost for a 7-day trip.

3.

ANSWER

C = 235t + 170, where C is total cost and t is time (in days); $1815

To write another equivalent equation, multiply each side by 0.5.

4x – 12y = 8

To write one equivalent equation, multiply each side by 2.

SOLUTION

Write two equations in standard form that are equivalent to 2x – 6y = 4.

EXAMPLE 1 Write equivalent equations in standard form

x – 3y = 2

SOLUTION

y – y1 = m(x – x1)

Calculate the slope.STEP 1

EXAMPLE 2 Write an equation from a graph

–3m =1 – (–2)

1 – 2=

3–1 =

Write an equation in point-slope form. Use (1, 1).

Write point-slope form.

y – 1 = –3(x – 1) Substitute 1 for y1, 3 for m

and 1 for x1.

Write an equation in standard form of the line shown.

STEP 2

Rewrite the equation in standard form.

EXAMPLE 2 Write an equation from a graph

3x + y = 4 Simplify. Collect variable terms on one side, constants on the other.

STEP 3

EXAMPLE 1GUIDED PRACTICE for Examples 1 and 2

Write two equations in standard form that are equivalent to x – y = 3.

1.

2x – 2y = 6, 3x – 3y = 9ANSWER

EXAMPLE 2 Write an equation from a graphGUIDED PRACTICE for Examples 1 and 2

Write an equation in standard form of the line through (3, –1) and (2, –3).

2.

–2x + y = –7ANSWER

SOLUTION

EXAMPLE 3 Write an equation of a line

Write an equation of the specified line.

The y-coordinate of the given point on the blue line is –4. This means that all points on the line have a y-coordinate of –4. An equation of the line is y = –4.

a.

The x-coordinate of the given point on the red line is 4. This means that all points on the line have an x-coordinate of 4. An equation of the line is x = 4.

b.

Blue linea. Red lineb.

Simplify.

Find the value of A. Substitute the coordinates of the given point for x and y in the equation. Solve for A.

STEP 1

SOLUTION

EXAMPLE 4

Find the missing coefficient in the equation of the line shown. Write the completed equation.

Ax + 3y = 2A(–1) + 3(0) = 2

–A = 2A = –2

Write equation.

Substitute –1 for x and 0 for y.

Divide by –1.

EXAMPLE 3EXAMPLE 4Complete an equation in standard form

EXAMPLE 4Complete an equation in standard form

Complete the equation.

–2x + 3y = 2 Substitute –2 for A.

STEP 2

Write equations of the horizontal and vertical lines that pass through the given point.

GUIDED PRACTICE for Examples 3 and 4

3. (–8, –9)

y = –9, x = –8ANSWER

GUIDED PRACTICE for Examples 3 and 4

4. (13, –5)

y = –5, x = 13ANSWER

Write equations of the horizontal and vertical lines that pass through the given point.

EXAMPLE 4Complete an equation in standard form

Find the missing coefficient in the equation of the line that passes through the given point. Write the completed equation.

EXAMPLE 3 Write an equation of a lineGUIDED PRACTICE for Examples 3 and 4

5. –4x + By = 7, (–1, 1)

ANSWER 3; –4x + 3y = 7

EXAMPLE 4Complete an equation in standard formEXAMPLE 3 Write an equation of a lineGUIDED PRACTICE for Examples 3 and 4

6. Ax + y = –3, (2, 11)

Find the missing coefficient in the equation of the line that passes through the given point. Write the completed equation.

ANSWER –7; –7x +y = –3

LIBRARY

EXAMPLE 5

Your class is taking a trip to the public library. You can travel in small and large vans. A small van holds 8 people and a large van holds 12 people. Your class could fill 15 small vans and 2 large vans.

b. Graph the equation from part (a).

c. List several possible combinations.

Solve a multi-step problem

Write an equation in standard form that models the possible combinations of small vans and large vans that your class could fill.

a.

SOLUTION

a. Write a verbal model. Then write an equation.

Because your class could fill 15 small vans and 2 large vans, use (15, 2) as the s- and l-values to substitute in the equation 8s + 12l = p to find the value of p.

8(15) + 12(2) = p Substitute 15 for s and 2 for l.144 = p Simplify.

Substitute 144 for p in the equation 8s + 12l = p.

EXAMPLE 5 Solve a multi-step problem

8 s l p12+ =

Substitute 0 for s.8(0) + 12l = 144

l = 12

Substitute 0 for l.

s = 188s + 12(0) = 144

ANSWER

The equation 8s + 12l = 144 models the possible combinations.

b. Find the intercepts of the graph.

EXAMPLE 5 Solve a multi-step problem

EXAMPLE 5 Solve a multi-step problem

Plot the points (0, 12) and (18, 0). Connect them with a line segment. For this problem only nonnegative whole-number values of s and l make sense.

The graph passes through (0, 12), (6, 8), (12, 4), and (18, 0). So, four possible combinations are 0 small and 12 large, 6 small and 8 large, 12 small and 4 large, 18 small and 0 large.

c.

EXAMPLE 5 Solve a multi-step problemEXAMPLE 5 Solve a multi-step problemGUIDED PRACTICE for Example 5

7. WHAT IF? In Example 5, suppose that 8 students decide not to go on the class trip. Write an equation that models the possible combinations of small and large vans that your class could fill. List several possible combinations.

8s + 12l = 136; 17 small, 0 large; 14 small, 2 large; 11 small, 4 large; 8 small, 6 large; 5 small, 8 large; 2 small, 10 large

ANSWER