### Use Linear Combination to Solve Systems of Equations and Inequalities

 Other Methods: Vocabulary, Possible Solutions Solver Graphically By Substitution With Calculator Using Determinants Using Matrices Linear Combination: Instructions   Example w/Opposite Coefficients Example w/Non Opposite Coefficients Problems w/Solutions

Linear Combination Means Combination of Lines

Linear combination means combination of lines.

Combination of lines means the addition of lines is part of the procedure required to solve the system.

So, to find the values of x and y, add lines together in a certain way.

Here's the instructions.

 Use Linear Combination to Solve Systems of Equations and Inequalities 1st: Rearrange the equations so terms line up as:   Ax + By = C 2nd: Multiply none, or one, or both equations by constant(s) so that the coefficients of one of the variables are opposites. 3rd: Add the two equations together to eliminate one of the variables. 4th: Solve. 5th: Use one of the equations and this value to solve for the other variable. 6th: State the solution and include values for both variables.

Look at an easy example first.
Solve a system in which coefficients of one variable are opposites.

1st: Rearrange the equations so terms line up as:   Ax + By = C
 Solve:2y + x = 64x - 2y = 12
becomes
 Solve: x + 2y = 6 4x - 2y = 12

2nd: Multiply none, or one, or both equations by constant(s)
so that the coefficients of one of the variables are opposites.

In this example, the above condition is already achieved.

The y coefficients are opposites, so, nothing more need be done.
 Solve: x + 2y = 6 4x - 2y = 12

3rd: Add the two equations together to eliminate one of the variables.
 Solve: x + 2y = 9 4x - 2y = -3
becomes
 Solve: x + 2y = 9 4x - 2y = -3 5x =6

4th: Solve.
 Solve: 5x = 6 5x/5 = 6/5 x = 6/5

5th: Use one of the equations and this value to solve for the other variable.
It doesn't matter which equation is used to do this.
 Solve: x + 2y = 9 6/5 + 2y = 9 2y = 9 - 6/5 2y = 39/5 2y/2 = (39/5)/2 y = 39/10

or
 Solve: 4x - 2y = -3 4(6/5) - 2y = -3 24/5 - 2y = -3 - 2y = -3 -24/5 - 2y = -39/5 - 2y/(-2) = (-39/5)/(-2) y = 39/10

5th ALTERNATE METHOD: Repeat steps 1 through 4 eliminating the other variable.
This alternate step avoids the messy substitution and computation with fractions as show in step 5 above.
 Solve: x + 2y = 9 4x - 2y = -3
becomes
 Solve: -4(x + 2y = 9) 4x - 2y = -3
becomes
 Solve: -4x -8y = -36 4x - 2y = -3 -10y =-39 -10y/(-10) =-39/(-10) y =39/10

6th: State the solution and include values for both variables.
If you use an ordered pair to do this, usually write in in alphabetical order.
 x=6/5 y=39/10
or
 (6/5, 39/10)

Solve A System in which Coefficients of One Variable Are Opposites.

The 2nd step: "Multiply none, or one, or both equations by constant(s) so that the coefficients of one of the variables are opposites." is the most fun!

Through it's simple arithmetic, a sophisticated combination of equations is reduced to a single easy equation.

It's a lot like finding a common demonimator in which the denominators are not erxactly the same but opposites.

 Solve: 6x +3y =2 5x +4y =8
when
multiplied
 Solve: 4(6x +3y =2) -3(5x +4y =8)

becomes
 Solve: 24x +12 =8 -15x -12y =-24
then
 9x =-16 x =-16/9

or

 Solve: 6x +3y =2 5x +4y =8
when
multiplied
 Solve: 5(6x +3y =2) -6(5x +4y =8)

becomes
 Solve: 30x +15 =10 -30x -24y =-48
then
 -9y =-38 y =-38/9

or

 Solve: 4x +5y =3 x +3y =6
when
multiplied
 Solve: 4x +5y =3 -4(x +3y =6)

becomes
 Solve: 4x +5y =3 -4x -12y =-24
then
 -7y =-21 y =-3

or

 Solve: 4x +5y =3 x +3y =6
when
multiplied
 Solve: -3(4x +5y =3) 5(x +3y =6)

becomes
 Solve: -12x -15y =-9 5x +15y =30
then
 -7x =21 y =-3

Or, these last examples which show how an inconsistent system and a dependent system look when using linear combination to solve the system.

 Solve: 4x +2y =1 4x +2y =6
when
multiplied
 Solve: 4x +2y =1 -(4x +2y =6)

becomes
 Solve: 4x +2y =1 -4x -2y =-24
then
 0 =-23 never true inconsistent systemno solution

or

 Solve: 4x +2y =1 -8x -4y =-2
when
multiplied
 Solve: 2(4x +2y =1) -8x -4y =-2

becomes
 Solve: 8x +4y =2 -8x -4y =-2
then
 0 =0 always true dependent systeminfinite number of solutions

 Solve the system. Mouseover the arrow to check answer. x - 2y = -2-3x - 6y = -6 y = -2x + 4y = -2x + 5 5x + 5y = 1010x + 10y = 12 y = 4x - 18y = 8x + 8 -2x - 2y = -43x - 8y = -12 -3x+3y=66x+9y=12 4x + 4y = 10x + y = 12 5x+15y=103x+9y=6