Warm-Up Exercises

Solve the linear system.

1. 4x – 3y = 152x – 3y = 9

2. –2x + y = –82x – 2y = 8

ANSWER (3, –1)

ANSWER (4, 0)

Warm-Up Exercises

3. You can row a canoe 10 miles upstream in 2.5 hours and 10 miles downstream in 2 hours. What is the average speed of the canoe in still water?

ANSWER 4.5 mi/h

Warm-Up Exercises

SOLUTION

EXAMPLE 1 Multiply one equation, then add

Solve the linear system:

6x + 5y = 19 Equation 1

2x + 3y = 5 Equation 2

STEP 1 Multiply Equation 2 by –3 so that the coefficients of x are opposites.

6x + 5y = 19

2x + 3y = 5

6x + 5y = 19

STEP 2 Add the equations. –4y = 4

–6x – 9y = –15

Warm-Up ExercisesEXAMPLE 1 Multiply one equation, then add

STEP 3

STEP 4

2x = 8

Write Equation 2.

2x + 3(–1) = 5 Substitute –1 for y.

2x + 3y = 5

x = 4

Multiply.

Subtract –3 from each side.

Solve for y.

Substitute –1 for y in either of the original equations and solve for x.

2x + (–3) = 5

Divide each side by 2.

y = –1

Warm-Up ExercisesEXAMPLE 1 Multiply one equation, then add

ANSWER The solution is (4, –1).

CHECK

Equation 2

2x + 3y = 5

Substitute 4 for x and –1 for y in each of the original equations.

Equation 1

6x + 5y = 19

6(4) + 5(–1) = 19?

2(4) + 3(–1) = 5?

19 = 19 5 = 5

Warm-Up ExercisesEXAMPLE 2 Multiply both equations, then subtract

Solve the linear system:

4x + 5y = 35 Equation 1

2y = 3x – 9 Equation 2

SOLUTION

STEP 1

4x + 5y = 35 Write Equation 1.

–3x + 2y = –9 Rewrite Equation 2.

Arrange the equations so that like terms are in columns.

Warm-Up ExercisesEXAMPLE 2 Multiply both equations, then subtract

STEP 2

4x + 5y = 35

–3x + 2y = –9

23x = 115STEP 3

STEP 4

8x + 10y = 70

–15x +10y = –45

Multiply Equation 1 by 2 and Equation 2 by 5 so that the coefficient of y in each equation is the least common multiple of 5 and 2, or 10.

Subtract: the equations.

x = 5Solve: for x.

Warm-Up ExercisesEXAMPLE 2 Multiply both equations, then subtract

STEP 5

4x + 5y = 35

4(5) + 5y = 35

y = 3

Write Equation 1.

Substitute 5 for x.

Solve for y.

ANSWER The solution is (5, 3).

Substitute 5 for x in either of the original equations and solve for y.

Warm-Up ExercisesEXAMPLE 2 Multiply both equations, then subtract

CHECK

4x + 5y = 35

ANSWER The solution is (5, 3).

Substitute 5 for x and 3 for y in each of the original equations.

4(5) + 5(3) = 35?

Equation 1 Equation 2

2y = 3x – 9

2(3) = 3(5) – 9?

35 = 35 6 = 6

Warm-Up ExercisesGUIDED PRACTICE for Examples 1 and 2

Solve the linear system using elimination.

–2x + 3y = –5

6x – 2y = 11.

ANSWER The solution is (–0.5, –2).

Warm-Up ExercisesGUIDED PRACTICE for Examples 1 and 2

3x + 10y = –3

2x + 5y = 32.

ANSWER The solution is (9, –3).

Solve the linear system using elimination.

Warm-Up ExercisesGUIDED PRACTICE for Examples 1 and 2

9y = 5x + 5

3x – 7y = 53.

Solve the linear system using elimination.

ANSWER The solution is (–10, –5).

Warm-Up ExercisesStandardized Test PracticeEXAMPLE 3

Darlene is making a quilt that has alternating stripes of regular quilting fabric and sateen fabric. She spends $76 on a total of 16 yards of the two fabrics at a fabric store. Which system of equations can be used to find the amount x (in yards) of regular quilting fabric and the amount y (in yards) of sateen fabric she purchased?

x + y = 16A x + y = 16 B

x + y = 16 Dx + y = 76 C

x + y = 76 4x + 6y = 76

6x + 4y = 764x + 6y = 16

Warm-Up ExercisesStandardized Test PracticeEXAMPLE 3

SOLUTION

Write a system of equations where x is the number of yards of regular quilting fabric purchased and y is the number of yards of sateen fabric purchased.

Equation 1: Amount of fabric

x + y = 16

Warm-Up ExercisesStandardized Test PracticeEXAMPLE 3

Equation 2: Cost of fabric

The system of equations is:x + y = 164x + 6y = 76

Equation 1

Equation 2

ANSWER

A DCBThe correct answer is B.

4 766+ =yx

Warm-Up ExercisesGUIDED PRACTICE for Example 3

SOCCER A sports equipment store is having a sale on soccer balls. A soccer coach purchases 10 soccer balls and 2 soccer ball bags for $155. Another soccer coach purchases 12 soccer balls and 3 soccer ball bags for $189. Find the cost of a soccer ball and the cost of a soccer ball bag.

4.

ANSWER

soccer ball $14.50, soccer ball bag: $5

Warm-Up ExercisesDaily Homework Quiz

Solve the linear system using elimination.

1. 8x + 3y = 12 –2x + y = 4

ANSWER (0, 4)

2. –3x + 2y = 7 5x – 4y = –15

ANSWER (1, 5)

3. –7x – 3y = 11 4x – 2y = 16

ANSWER (1, –6)

Warm-Up ExercisesDaily Homework Quiz

A recreation center charges nonmembers $3 to use the pool and $5 to use the basketball courts. A person pays $42 to use the recreation facilities 12 times. How many times did the person use the pool.

4.

ANSWER 9 times