Solving Systems by Graphing and Using Substitution

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Solving Systems by Graphing and Using Substitution
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System of Linear Equations

• Two or more linear equations graphed on the same
  grid or pertaining to the same situation.

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Solution of a System

• Any point common to all the lines is a solution.
• Any ordered pair that makes all the equations
  true is a solution of a system.

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Solving Systems by Graphing

• Make sure both/all the equations are in the
  slope-intercept form (y ).
• Enter the equations in y on the graphing
  calculator.
• Make sure your plot is OFF.
• Graph (you may have to zoom 6 to get it
  centered).
• If you cannot see the point of intersection, zoom
  out.

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Solving Systems by Graphing

• When you can see the point of intersection, go to
  Calc (2nd Trace)
• Go to 5 intersect
• Press enter, enter, enter.
• The two numbers at the bottom of the screen are
  the ordered pair that is the solution.
• Check by replacing the x and y with the
  appropriate numbers in both equations. The
  numbers should make BOTH equations true.

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Example 1

• Find the solution of this system by graphing
• y 2x 3
• y x 1
• Answer (2, 1)
• Check. Replace x with 2 and y with 1 in both
  equations
• 1 2(2) 3 True
• 1 2 1 True

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Example 2

• Solve y x 5
• y -4x
• Answer (-1, 4)
• Check Replace x with 1 and y with 4 in the
  first and second equations
• 4 -1 5 True
• 4 -1 x 4 True

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Example 3

• Solve y 3x 4
• y 3x 6
• Answer The lines have the same slope with
  different y-intercepts so they are parallel.
  There is no intersection, so there is NO
  SOLUTION.

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Example 4

• Solve y 2x 1
• -2x y 1
• Put the second equation in the y form, then
  graph.
• Answer When you put the second equation in the
  y form, they are the same equation. They
  graph right on top of each other (or coincide).
  Therefore, the answer is y 2x 1.

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Solving Systems Using Substitution
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Substitution

• Sometimes you dont have a graphing calculator to
  use.
• Substitution allows you to solve a one-variable
  equation.
• One of the equations should be in either y or x
  form.
• Replace that variable in the other equation with
  the stuff on the right side.

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Example 5

• Solve y 2x and 7x y 15
• Replace the y in the second equation with what y
  is equal to in the first equation
• 7x (2x) 15
• Simplify 5x 15
• Solve x 3
• Now, replace x in either equation with 3 to find
  y y 2(3)
• Y 6 so the solution is (3, 6)
• Check in the second equation 7(3) (6) 15
• 21 6 15 is true so the solution is (3, 6)

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Example 6

• Solve y 4x and 2x - 0.5y 0
• Replace the y in the second equation with the
  4x
• 2x 0.5(4x) 0
• Simplify 2x 2x 0
• 0 0 A true statement, so there are infinitely
  many solutions.
• So, as the solution, you write y 4x.

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Example 7

• Solve y 5 3x and 6x 2y 1
• Replace the y in the second equation with what y
  is equal to (5 3x)
• 6x 2(5 3x) 1
• Distribute the 2 6x 10 6x 1
• Simplify 10 1 Not a true statement so there
  is no solution.

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Example 8

• Solve y 2x 3 and x 2y 3
• Replace y with 2x 3 in the second equation.
• X 2(2x 3) 3
• Distribute the 2 x 4x 6 3
• Simplify -3x 6 3
• Solve -3x 9 x -3
• Replace x with 3 in the first equation y
  2(-3) 3 y -3
• So, the solution is (-3, -3) (Check it in the 2nd
  equation too.)