Solving Systems of Linear Equations by Substitution

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Solving Systems of Linear Equations by
Substitution
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Steps to Solve by substitution

1. Solve ONE of the equations for one of its
   variables
2. Substitute the expression for the variable found
   in Step 1 into the other equation
3. Solve the equation from step 2 to find the value
   of the variable.
4. Using the value in Step 3 substitute the value
   into one of the equations to solve for the other
   variable
5. Check your answer!

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Example 2x 3y 13 and x y 4

• Using y 1 use either equation to solve for X
• X 1 4 5
• 2x 3(1) 13
• 2X 3 13
• 2X 10
• X 5
• Check 5 1 4
• 2(5) 3(1) 13
• Yes so therefore, (5,1) is our solution!

• Use x y 4
• x y 4 - expression
• 2(y 4 ) 3y 13
• 2y 8 3y 13
• 5y 8 13
• -8 -8
• 5y 5
• Y 1

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Example 3x y 5 and 3x 2y -7

• 3. 3x -2(-3x 5) -7
• 3x 6x 10 -7
• 9x 10 -7
• 9x 3
• X 1/3
• 3(1/3) y 5
• 1 y 5
• Y 4
• 3(1/3) 2(4) -7
• 1 8 -7
• Answer (1/3, 4)

• Solve an equation for one of the variables
• 3x y 5
• -3x -3x
• Y -3x 5
• 2. 3x -2(-3x 5) -7

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Example -x 3y 6 and 3y x 6

• 1). 3y x 6
• 3 3
• Y (1/3)x 2
• 2). -x 3((1/3)x 2) 6
• 3). -x x 6 6
• 6 6
• Infinite amount of solutions!

• Remember what type of systems have infinite
  amount of solutions!
• Same slope and same y-intercept!

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Example 2x -3y 6 and -4x 6y 5

• 2x 3y 6
• 3y 3y
• 2x 3y 6
• 2 2
• X (3/2)y 3
• 2. -4((3/2)y 3) 6y 5
• -6y -12 6y 5
• -12 5
• No solution!

• What type of systems have no solution?
• Parallel lines!