How would you solve the following system of linear equations?
x – y = 5
2x + y = 13
There are two primary approaches for solving systems of linear equations:
- Substitution Method
- 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 (2x + 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. When we do so, we get 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 . When we do so, we get 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 2x + y = 13), the y’s cancel out (i.e., they are eliminated), leaving us with: 3x = 18.
From here, when we divide both sides by 3, we get: x = 6, and from here we can find the value of y . We get 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 will get the job done on the GMAT, 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:
5x – 2y = 7
3x + 2y = 17
To use the Substitution Method here, we’d have to deal with messy fractions. For example, if we take the equation 5x – 2y = 7 and solve for x , we get x = (2/5)y + 7/5. Then, when we take the second equation (3x + 2y = 17) and replace x with (2/5)y + 7/5, we get: 3[(2/5)y + 7/5] + 2y = 17. Yikes!!
Alternatively, we can use the Elimination Method and add the two original equations (5x – 2y = 7 and 3x + 2y = 17). When we do this, the y ’s cancel, leaving us with: 8x = 24, which means x = 3. No messy fractions.
It has been my experience that many GMAT 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 not only learn the Elimination Method but also make it your number one solution strategy. I’d almost go as far as suggesting that you drop the Substitution Method altogether from your repertoire, but there’s probably an obscure GMAT question out there that is better handled using the Substitution Method.
In conclusion, if you still feel that the Substitution Method is superior, try using it to solve the following question:
If 4x – 7y = 9, and 3x – 8y = 4, what is the value of x + y ?
To view the solution to the above question, see my last article, The Reasonable Test-Maker.