3'2 Solving Linear Systems Algebraically

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3.2 Solving Linear Systems Algebraically

• Goal Use Algebraic Methods to Solve Linear
  System.

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The Substitution Method

• Step 1 Solve 1 of the equations for 1 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 revised equation from Step 1 and Solve.

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Example 1 Using the Substitution Method

• 3x 4y -4
• x 2y 2
• x 2 2y (solved 2nd equation for x)
• 3(2 2y) 4y -4 (substitute in for x)
• 6 6y 4y -4
• -2y -10
• y 5
• x 2 2(5) 2 10 -8 (substitute in for
  y)
• So, (x, y) (-8, 5) is the solution to the
  linear system

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Solve the following System using the Substitution
Method

• 6x y -2
• 4x 3y 17

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The Combination Method

• Step 1 Multiply 1 or both equations by a
  constant to obtain coefficients that only differ
  in sign for one of the variables.
• Step 2 Add the revised equations from Step 1
  which will eliminate one of the variables. Solve
  for the remaining variable.
• Step 3 Substitute the value obtained in Step 2
  into either of the original equations and solve
  for the other variable.

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Example 2 Using the Combination Method

• 2x - 4y 13
• 4x - 5y 8
• -2(2x 4y 13) -4x 8y -26
  (multiplied 1st equation by -2)
• -4x 8y -26 (add the equations)
• 4x 5y 8
• 3y -18
• y -6
• 3. 2x 4(-6) 13 (substitute in for y)
• 2x 24 13
• 2x -11
• x

So, (x, y) (-6, -11/2) is the solution to the
linear system
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Solutions From Homework Pg. 152 11-20 every 3rd
problem

• 11. (x, y) (4, -1)
• 14. (x, y) (4.4 or 22/5, -6)
• 17. (x, y) (-2.4 or 12/5, 10.2 or 102/10)
• 20. (x, y) (8.14 or 57/7, -1.29 or 9/7)

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Solve the following System using the Combination
Method

• x 2y 1
• -3x 4y 1

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Solve the following System using the Combination
Method

• 7x 12y -22
• -5x 8y 14

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Linear Systems with Many or No Solution

• x 2y 3 b. 6x 10y 12
• 2x 4y 7 -15x 25y -30

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Key concept
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Using a Linear System as a Model

• A caterer is planning a party for 64 people. The
  customer has 150 to spend. A 39 pan of pasta
  feeds 14 people and a 12 sandwich tray feeds 6
  people. How many pans of pasta and how many
  sandwich trays should the caterer make?

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Homework

• Pg. 152 20-44 every 3rd problem
• 23 32 use the Combination Method
• 32 44 use Substitution, Combination, or
  Graphing Method

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Solutions

• 23) (-11/3 or 3.666, -1
• 26) (-1, -1)
• 29) (1/3, 1)
• 32) Infinite
• 35) (-5, -2)
• 38) (-6, -7)
• 41) (-69/11 or 6.27, 65/11 or 5.9)
• 44) (-5,-5)