Use Linear Combination to Solve Systems of Equations and Inequalities
Other Methods: 
Vocabulary, Possible Solutions 
Solver 

Graphically
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Using Determinants
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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 = 6 4x  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.

 or

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  

 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  

 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  

 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  
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 system no 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 system infinite number of solutions 

Solve the system. Mouseover the arrow to check answer. 
x  2y = 2 3x  6y = 6

y = 2x + 4 y = 2x + 5

5x + 5y = 10 10x + 10y = 12

y = 4x  1 8y = 8x + 8

2x  2y = 4 3x  8y = 12

3x+3y=6 6x+9y=12

4x + 4y = 10 x + y = 12

5x+15y=10 3x+9y=6







 ©
2008, A. Azzolino
 www.mathnstuff.com/math/algebra/asystem.htm
