3 2 solving linear systems algebraically
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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 2 solving linear systems algebraically

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

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.


Example 1 substitution

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)


Your turn to try

Your turn to try.

3x – y = 13

2x + 2y = -10


The elimination method

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.


Example 2 elimination multiplying 1 equation

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

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)


Example 3 elimination multiplying 2 equations

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


Continued1

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

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


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