Advanced Algebra / Trigonometry

Section 3-2Solving Systems of Equations

Algebraically

Target Goals

1) Solve systems of linear equations by using substitution.

2) Solve systems of linear equations by using elimination.

Solving Systems of Equations by Substitution

Use substitution to solve the system of equations.

5 3 23

2 7

x y

x y

Solve 2nd equation for . y

Step 1

Example 1

2 7x y 2 7y x

Step 2

5 3 232 7xx 5 6 21 23x x

11 21 23x 11 44x

4x

Step 3

2 4 7y 8 7y

1y

Solution

4, 1

Example 2

Example 3

no solution What does look like?Use substitution to solve the system of equations.

Step 1

Step 2 Step 35 15 24

2 6 28

x y

x y

Solve 2nd equation for x2 6 28x y 2 6 28x y

3 14x y

3 1 15 2445 y y 15 70 15 24y y

70 24 The variable "disappears" and youhave a "false statement".In this case the system has "no solution".

infinitely many solutionsWhat does look like?IMSUse substitution to solve the system of equations.9 3 18

3 6

y x

y x

Step 1

Step 2 Step 3

Solve 2nd equation for x3 6y x

3 6x y 3 6x y

9 3 183 6yy 9 9 18 18y y

18 18The variable "disappears" and youhave a "true statement".In this case the system has IMS

IMS

Exit Slip

.A

.B

Target Goal: Solve system of linear equations by substitution.

.C

.D

7 11

5 4 23

x y

x y

Use substitution to solve the system of equations.

3,2

3, 2

3, 2

.A

.B

Target Goal: Solve system of linear equations by substitution.

.C

.D

5

3 3 9

y x

y x

Use substitution to solve the system of equations.

6,1

1,4

1,2

Solving Systems of Equations by Elimination

Example 4Use elimination to solve the system of equations.

5 3 23

2 7

x y

x y

Step 1

Step 2 Step 3

Multiply 2nd equation by 3

2 37x y

6

5

3

3

21

23

x y

x y

11 44x 4x

5 4 3 23y 20 3 23y

3 3y 1y

Solution

4, 1

Example 5

Example 6

no solution What does look like?Use elimination to solve the system of equations.

Step 1

Step 2 Step 35 15 24

2 6 28

x y

x y

infinitely many solutionsWhat does look like?IMSUse elimination to solve the system of equations.9 3 18

3 6

y x

y x

Step 1

Step 2 Step 3

Multiply 1st equation by 2 andMultiply 2nd equation by 5

24 25 15x y

2 28 56x y

10 30 48

10 30 140

x y

x y

0 92

The variable "disappears" and youhave a "false statement".In this case the system has "no solution".

Multiply 2nd equation by 3

9

9

3

3

18

18

y x

y x

3 6 3y x 0 0

The variable "disappears" and youhave a "true statement".In this case the system has IMS

IMS

Writing Systems of Linear EquationsExample 7At a park, there are 38 people playing tennis. Some are playing doubles,and some are playing singles. There are 13 matches in progress. A doublesmatch requires 4 players, and a singles match requires 2 players. How manymatches of each kind are in progress?

Let number of doubles matches.x Let the number of singles matches.y

13x y 4 2 38x y

Step 1

Step 2 Step 3

Multiply the 1st equation by 2.

213x y

2 2 26x y 4 2 38x y

2 12x 6x

4 6 2 38y 24 2 38y

2 14y 7y

6 doubles and 7 singles matches

Exit Slip

.A

.B

.C

.D

Target Goal: Use elimination to solve the system of equations.

4 3 2

4 2 12

x y

x y

8, 10

2, 2

10,14

.A

.B

.C

.D

Target Goal: Use elimination to solve the system of equations.

8 3 12

32 12 48

x y

x y

3, 4

0,4

1, 1

End

Advanced Algebra / Trigonometry

Section 3-2Solving Systems of Equations

Algebraically

Target Goals

1) Solve systems of linear equations by using substitution.

2) Solve systems of linear equations by using elimination.

Solving Systems of Equations by Substitution

Use substitution to solve the system of equations.

5 3 23

2 7

x y

x y

Step 1

Example 1

Step 2 Step 3

Example 2

Example 3

no solution What does look like?Use substitution to solve the system of equations.

Step 1

Step 2 Step 35 15 24

2 6 28

x y

x y

infinitely many solutionsWhat does look like?IMSUse substitution to solve the system of equations.9 3 18

3 6

y x

y x

Step 1

Step 2 Step 3

Exit Slip

.A

.B

Target Goal: Solve system of linear equations by substitution.

.C

.D

7 11

5 4 23

x y

x y

Use substitution to solve the system of equations.

3,2

3, 2

3, 2

.A

.B

Target Goal: Solve system of linear equations by substitution.

.C

.D

5

3 3 9

y x

y x

Use substitution to solve the system of equations.

6,1

1,4

1,2

Solving Systems of Equations by Elimination

Example 4Use elimination to solve the system of equations.

5 3 23

2 7

x y

x y

Step 1

Step 2 Step 3

Example 5

Example 6

no solution What does look like?Use elimination to solve the system of equations.

Step 1

Step 2 Step 35 15 24

2 6 28

x y

x y

infinitely many solutionsWhat does look like?IMSUse elimination to solve the system of equations.9 3 18

3 6

y x

y x

Step 1

Step 2 Step 3

Writing Systems of Linear EquationsExample 7At a park, there are 38 people playing tennis. Some are playing doubles,and some are playing singles. There are 13 matches in progress. A doublesmatch requires 4 players, and a singles match requires 2 players. How manymatches of each kind are in progress?

Let number of doubles matches.x Let the number of singles matches.y

Step 1

Step 2 Step 3

Exit Slip

.A

.B

.C

.D

Target Goal: Use elimination to solve the system of equations.

4 3 2

4 2 12

x y

x y

8, 10

2, 2

10,14

.A

.B

.C

.D

Target Goal: Use elimination to solve the system of equations.

8 3 12

32 12 48

x y

x y

3, 4

0,4

1, 1