write linear equations in standard form you will write equations in standard form how do you write...

Write Linear Equations in Standard Form

• You will write equations in standard form

• How do you write equations in standard form?

You will learn how to answer this question by rewriting equations.

Warm-Up Exercises

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

Warm-Up Exercises

SOLUTION

y = mx +b

Calculate the slope.STEP 1

EXAMPLE 2 Write an equation from a graph

–3m =1 – (–2)

1 – 2=

3–1 =

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

1 = –3( 1) + b

Write an equation in standard form of the line shown.

STEP 2

b= 4

y= -3x + 4

Warm-Up Exercises

Rewrite the equation in standard form.

EXAMPLE 2 Write an equation from a graph

3x + y = 4

STEP 3

y= -3x + 4

Warm-Up ExercisesEXAMPLE 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

Warm-Up ExercisesEXAMPLE 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

Warm-Up Exercises

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.

Warm-Up Exercises

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

Warm-Up ExercisesEXAMPLE 4Complete an equation in standard form

Complete the equation.

–2x + 3y = 2 Substitute –2 for A.

STEP 2

Warm-Up Exercises

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

Warm-Up ExercisesGUIDED 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.

Warm-Up ExercisesEXAMPLE 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

Warm-Up ExercisesEXAMPLE 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

Warm-Up Exercises

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.

Warm-Up Exercises

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+ =

Warm-Up Exercises

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

Warm-Up ExercisesEXAMPLE 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.

Warm-Up ExercisesDaily Homework Quiz

Write an equation in standard form of the line that passes through the given point and has the given slope m or that passes through the two given points.

ANSWER 2x + y = –4

1.

(1, –6), m = –2

2. (–4, –3), (2, 9)

ANSWER –2x + y = 5

Warm-Up ExercisesDaily Homework Quiz

ANSWER

8p + 6c = 96; 16 h kayak and 0 h paddleboat; 12 h paddleboat and 0 h kayak; 6 h paddleboat and 8 h kayak.

3.

You have $96 to spend on campground activities. You can rent a paddleboat for $8 per hour and a kayak for $6 per hour. Write an equation in standard form that models the possible hourly combinations of activities you can afford. List three possible combinations.

• You will write equations in standard form

• How do you write equations in standard form?

Ax + By = C is the standard form of an equation where A, B, and C are real numbers, and A and B are not both 0.

Use the slope formula to find the slope. Write the equation in y = mx + b form. Collect the variable terms on one side of the equation and constants on the other to rewrite the equation in standard form.