The student will be able to:

1
Objective

• The student will be able to
• solve systems of equations using elimination with
  multiplication.

2
Solving Systems of Equations

• So far, we have solved systems using graphing,
  substitution, and elimination.
• This goes one step further and show how to use
  ELIMINATION with multiplication.
• What happens when the coefficients are not the
  same?
• We multiply the equations to make them the same!
  Youll see

3
Solving a system of equations by elimination
using multiplication.

• Step 1 Put the equations in Standard Form.

Standard Form Ax By C
Step 2 Determine which variable to eliminate.
Look for variables that have the same coefficient.
Step 3 Multiply the equations and solve.
Solve for the variable.
Step 4 Plug back in to find the other variable.
Substitute the value of the variable into the
equation.
Step 5 Check your solution.
Substitute your ordered pair into BOTH equations.
4
1) Solve the system using elimination.
2x 2y 6 3x y 5
Multiply the bottom equation by 2 2x 2y
6 (2)(3x y 5)
2x 2y 6 () 6x 2y 10
8x 16 x 2
2(2) 2y 6 4 2y 6 2y 2 y 1
(2, 1)
5
2) Solve the system using elimination.

• x 4y 7
• 4x 3y 9

Multiply the top equation by -4 (-4)(x 4y
7) 4x 3y 9
-4x 16y -28 () 4x 3y 9
-19y -19 y 1
4x 3y 9 4x 3(1) 9 4x 3 9 4x 12 x 3
(3, 1)
6
Which variable is easier to eliminate?
3x y 4 4x 4y 6

• y is easier to eliminate.
• Multiply the 1st equation by (-4)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30 31 32
7
3) Solve the system using elimination.

• 3x 4y -1
• 4x 3y 7

Multiply both equations (3)(3x 4y
-1) (4)(4x 3y 7)

9x 12y -3 () 16x 12y 28
25x 25 x 1
3(1) 4y -1 3 4y -1 4y -4 y -1
(1, -1)
8
4) What is the best number to multiply the top
equation by to eliminate the xs?
3x y 4 6x 4y 6

• Multiply the 1st equation by (-2)
• (-2)(3x y 4) ? -6x -2y -8
• 6x 4y 6
• 2y -2
• y 1

6x 4y 6 6x 4(1) 6 6x 2 x
2/6 x 1/3
(1/3, 1)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30 31 32
9
5) Solve using elimination.
2x 3y 1 x 2y -3

• Multiply 2nd equation by (-2) to eliminate the
  xs.
• (-2)(x 2y -3) ? -2x 4y 6
• 1st equation ? 2x 3y 1
• -7y 7
• y -1

2x 3y 1 2x 3(-1) 1 2x 3 1 2x
-2 x -1
(-1, -1)
10
5) Solve using elimination.

• 2(4x y 6 )
• -8x 2y 13
• 8x 2y 12
• -8x 2y 13
• 0 25

Add down to eliminate x. But look what happens,
y is eliminated too. We now have a false
statement, thus the system has no solution, it is
inconsistent.
11
6) Elimination
Solve the system using elimination. 5x 2y
15 3x 8y 37 20x 8y 60 3x 8y
37 23x 23 x 1
(4)
Since neither variable will drop out if the
equations are added together. Multiply one or
both of the equations by a constant to make one
of the variables have the same number with
opposite signs.
The best choice is to multiply the top equation
by 4 since only one equation would have to be
multiplied. Also, the signs on the y-terms
are already opposites.
12
6) Elimination
x 1
3x 8y 37 (second equation) 3(1) 8y 37 3
8y 37 8y 40 y 5 The solution
is (1, 5)
To find the second variable, substitute in any
equation that contains two variables.
13
7) Elimination
Solve the system using elimination. 4x 3y
8 3x 5y 23 20x 15y 40 9x 15y
69 29x 29 x 1
(5)
For this system, we must multiply both equations
by a different constant in order to make one of
the variables drop out.
(3)
It would work to multiply the top equation by 3
and the bottom equation by 4 OR to multiply the
top equation by 5 and the bottom equation by 3.
14
7) Elimination
x 1
4x 3y 8 4(1) 3y 8 4 3y 8 3y
12 y 4 The solution is (1, 4)
15
8) Elimination Method

• Solve the system.
• To eliminate x, multiply equation (1) by 2 and
• equation (2) by 3 and add the resulting equations.

(1) (2)
16
8) Elimination Method

• Substitute 2 for y in (1) or (2).
• The solution is (3, 2)

17
9) Solving an Inconsistent System

• Solve the system
• Eliminate x by multiplying (1) by 2 and
• adding the result to (2).
• Solution set is ?.

(1) (2)
Inconsistent System
18
10) Solving a System with Dependent Equations

• Solve the system.
• Eliminate x by multiplying (1) by 2 and adding
  the result to (2).
• Each equation is a solution of the other.
• Infinite solutions.

(1) (2)