# 3.2 Solving Linear Systems Algebraically - PowerPoint PPT Presentation

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3.2 Solving Linear Systems Algebraically . I can solve a two variable system by substitution. I can solve a two variable system by elimination. . The Substitution Method. Step 1: Solve one of the equations for one of its variables. Step 2:

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3.2 Solving Linear Systems Algebraically

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## 3.2Solving Linear Systems Algebraically

I can solve a two variable system by substitution.

I can solve a two variable system by elimination.

### The Substitution Method

• Step 1:

• Solve one of the equations for one of its variables.

• Step 2:

• Substitute the expression from Step 1 into the other equation and solve for the other variable.

• Step 3:

• Substitute the value from Step 2 into the equation from Step 1 and solve.

Which equation is easiest to get a variable?

### Example 1: Substitution

3x + 4y = -4

x + 2y = 2

• Step 1: solve for a variable

x + 2y = 2

-2y -2y

x = 2 – 2y

• Step 3: Substitute the value into Step 1

x = 2 – 2(5)

x = 2 – 10

x = -8

• Step 2: Substitute into other equation

3x + 4y = -4

3(2 – 2y) +4y = -4

6 – 6y + 4y = -4

6 – 2y = -4

-6 -6

-2y = -10

-2 -2

y = 5

(-8,5)

3x – y = 13

2x + 2y = -10

### The Elimination Method

• Step 1:

• Multiply one or both of the equations by a constant (#) to get both a + and – coefficient for a variable.

• Step 2:

• Add the revised equation(s) from Step 1. By combining like terms, one of your variables will eliminate. Solve for the remaining variable.

• Step 3:

• Substitute the value from Step 2 into either original equation and solve for the other variable.

Which variables are multiples of each other?

### Example 2: Elimination (Multiplying 1 Equation)

2x – 4y = 13

4x – 5y = 8

• Step 1: Multiply to get a + and – variable.

-2(2x – 4y = 13)

-4x + 8y = -26

• Step 2: Add the revised equation and combine like terms.

-4x + 8y = -26

4x – 5y = 8

3y = -18

3 3

y = -6

### Continued….

2x – 4y = 13

4x – 5y = 8

• Step 3: Substitute into an original equation to solve for the other variable.

• y = -6

2x – 4y = 13

2x – 4(-6) = 13

2x + 24 = 13

-24 -24

2x = -11

2 2

x =

(-11/2,-6)

Choose a variable and multiply it by the variable in the other equation.

### Example 3: Elimination (Multiplying 2 Equations)

7x – 12y = -22

-5x + 8y = 14

Step 1: Multiply to get the same + and – variable.

5(7x – 12y = -22)

35x – 60y = -110

7(-5x + 8y = 14)

-35x + 56y = 98

Step 2: Combine like terms

35x – 60y = -110

-35x + 56y = 98

-4y = -12

-4 -4

y = 3

### Continued…

7x – 12y = -22

-5x + 8y = 14

y = 3

Step 3: Substitute into an original equation

7x – 12y = -22

7x – 12(3) = -22

7x – 36 = -22

+36 +36

7x = 14

7 7

x = 2

(2,3)

### Table Partner Classwork

• Textbook

• Pg 153

• 35-40

• When you are finished, check with me.

• Start on homework.

• Homework

Solve the system using a chosen method.

• x + 5y = 33

4x + 3y = 13

• -2x + 3y = -13

6x + 2y = 28