Solving Linear Systems By Combining Equations

Solve the following linear systems using the combination technique.  Hints are provided.

(solutions are provided at the bottom)

When approaching the following problems  you should look for a way to cancel out one of the variables.  Often getting variables to cancel involves multiplying an equation by a constant factor in order to make a coefficient in that equation equal the negative of the coefficient for the same variable in the other equation.

For example, if our two equations are

2x - y = 10

3x + 2y = 14

then we must choose a convenient factor.  Notice that in the first equation we have a -y and in the second we have 2y.  In this case, multiplying the first equation by 2 will give us a -2y in the first equation and a 2y in the second. Now the y terms will cancel.

4x - 2y = 20   (we multiplied 2 times 2x - y = 10)

3x + 2y = 14

adding these up gives us 7x + 0y  = 34.  

To recap, you just need to get one of the variables to cancel out.  Choose a factor that will make one coefficient in one equation equal to the negative of that same coefficient in the other equation.  Now try these exercises, good luck.

1)

-x + y = 10

x - 2y = 12

(hint: just try adding the two together)

2)

3x + 2y = 5

6x - y = 20

(hint: multiply one of the equations by two before combining them)

3)

2x  + 5y = 4

3x + 2y = -5

(hint: multiply the top equation by 3 and the bottom equation by -2)

SOLUTIONS

1)  (x,y) = (-32,-22)

2) (x,y) = (3,-2)

3) (x,y) = (-3,2)