## The Power of Elimination

How would you solve the system of linear equations below?

** x – y = 5**

** 2 x + y = 13**

There are two primary approaches for solving systems of linear equations:

1) Substitution Method

2) Elimination Method

## The Substitution Method

With this method, we take one of the equations and solve for a certain variable. For example, we might take *x* – *y* = 5 and add *y* to both sides to get *x *= *y* + 5.

Then we take the second equation (2*x* + *y* = 13) and replace *x *with *y* + 5 to get: 2(*y* + 5)+ *y* = 13

From here, we have an equation we can solve for *y*: *y* = 1

Now that we know the value of *y*, we can take one of the equations and replace *y* with 1 to find the value of *x*: *x *= 6

So, the solution is* x *= 6 and *y* = 1.

## The Elimination Method

With this method, we notice that, if we add the two original equations (*x* – *y* = 5 and 2*x* + *y* = 13), the *y*’s cancel out (i.e., they are eliminated), leaving us with: 3*x* = 18.

From here, when we divide both sides by 3, we get: *x* = 6, and from here we can find the value of *y*: *y *= 1.

Okay, so that’s how the two methods work. What’s my point?

The point I want to make is that, although both methods get the job done, the Elimination method is superior to the Substitution method. And by “superior,” I mean “faster.”

First, the Elimination method can often help us avoid using fractions. Consider this system:

** 5 x – 2y = 7**

** 3 x + 2y = 17**

To use the Substitution method here, we’d have to deal with messy fractions. For example, if we take the equation 5*x* – 2*y* = 7 and solve for *x*, we get *x* = (2/5)*y* + 7/5. Then when we take the second equation (3*x* + 2*y* = 17) and replace *x* with (2/5)*y* + 7/5, we get: 3[(2/5)*y* + 7/5] + 2*y* = 17. Yikes!!

Alternatively, we can use the Elimination method and add the two original equations (5*x* – 2*y* = 7 and 3*x* + 2*y* = 17). When we do this, the *y*’s cancel, leaving us with: 8*x* = 24, which means *x* = 3. No messy fractions.

It has been my experience that many students rely solely on the Substitution method to solve systems of equations, and this can potentially eat up a lot of time on test day. So, be sure to learn the Elimination method soon. In fact, if I were you, I’d drop the Substitution method from my repertoire; it isn’t very useful.

If you still believe that the Substitution method is just as good as the Elimination method, try solving this question using the Substitution method.

**If 5x – 8y = 11, and 4x – 9y = 4, what is the value of x + y?**

**3****4****5****6****7**

To view the solution to the above question, see my post, The Reasonable Test-Maker.