Solving Systems of Equations by Elimination or Combination

1
Solving Systems of Equationsby Elimination or
Combination

• Find the point (x, y) at which the lines, if
  graphed, intersect.
• That point (x, y) if subbed in to either
  equation, would make a true statement.
• That point (x, y) is the solution to the system
  of equations
• There may be 1 solution, 0 solutions, or an
  infinite number of solutions. Why?

2
Eliminate What?

• Eliminate either x or y
• Choose the easiest to eliminate
• Once eliminated, solve for the other variable
• Now, you have a value for either x or y
• Sub that value in to one of the original
  equations
• Solve for the other variable
• Rewrite as a point (x, y)

3
Eliminate How?

• Put into Standard Form (no fractions, x y on
  left side of equation)
• Make sure like terms are lined up
• Determine which variable to eliminate
• Rename the equation by multiplying it by the
  necessary number
• Combine like terms
• Solve for the remaining variables value

4
Example 1
ThinkThinkThink Which variable to eliminate?
Solve the linear system
Should 2x become -6x? Should 3y become -5y?
SOLUTION
SOLUTION
SOLUTION
STEP 1 Multiply Equation 2 by -3 so that the
coefficients of x are opposites and x will be
eliminated.
STEP 2 Combine Terms.
STEP 3 Solve for y.
STEP 4 Sub in the value of y that you found and
now, find x! Your answer is (x,y)
5
Example 1
Solve the linear system
SOLUTION
SOLUTION
SOLUTION
STEP 1 Multiply Equation 2 by -3 so that the
coefficients of x are opposites and x will be
eliminated.


6x 5y 19

6x 5y 19

6x 5y 19






-3 2x 3y 5


6x 9y 15


6x 9y 15

4y
STEP 2 Combine Terms.
4


4y
6
Example 1
Multiply one equation, then add
7
Example 1
Multiply one equation, then add
CHECK Substitute 4 for x and 1 for y in each
of the original equations.

8
Example 2
Multiply both equations, then subtract
Solve the linear system
9
Example 2
Multiply both equations, then subtract
ThinkThinkThink Which variable to eliminate?
Should -3x become -4x? Should 2y become -5y?
10
Example 2
Multiply both equations, then subtract
STEP 5 Substitute 5 for x in either of the
original equations and solve for y.
11
Example 2
Multiply both equations, then subtract
12
for Examples 1 and 2
Solve the linear system using elimination.
13
for Examples 1 and 2
Solve the linear system using elimination.