by ejkiv

July 18, 2015

GMAT Systems of equations will come into play when you have multiple variables and multiple equations.

**The general rule is that in order to solve for a linear equation with N variables you will need to have N different equations to solve for these variables.** This rule is not hard and fast, but it is a good place to start.
When looking at linear equations, there is only going to be 1 solution for each variable. This is different in quadratic equations, which can have multiple solutions. Again, we will cover this in a couple of chapters, but it is important to acknowledge the difference before we continue moving.
There are two ways in which you are going to solve: combination and substitution. These methods ultimately get you to the same place; however, in certain instances one will be easier to use than the other.
Let’s look at combination first (as it is the one I prefer). In combination you are looking to add or subtract equations to isolate one of the variables.

In this system, we can simply add the two equations together to eliminate the y variable, thus isolating the x variable.

Here you will be able to solve x = 2. Once you have the x-value, simply plug it into either of the equations to solve for y (you will get the same answer). Either way, the answer comes to y = 0. And you are done!
Since we added before, let’s look at a more complicated example with subtraction:

While the previous example was straightforward, there is no way to easily combine these equations. First, you have to get the coefficients of one of the variables to be identical in both of the equations. In this case, I am going to multiply the top equation by 3. This will give me 3x + 6y = 12. The reason I chose this route as opposed to combining the y variables is that I did not want to have negative numbers. Remember, we are trying to keep stuff simple, and the more difficulty you introduce, the more likely you are to make a mistake. Here are the next steps of the calculation when I subtract the second equation from the new first equation:

After this, you can substitute your answer for y back into one of the equations:

Either way will work, just get comfortable solving in different ways so you can quickly decide which is best when approaching a particular problem.
The other way that you can solve for a system of equations is by substitution. The basic method here is to solve for one variable in terms of the other and then substitute the expression for the variable in the other equation. Here is an example:

In order to do this, we must first isolated a variable to substitute. In this example, the first equation can be rearranged to y = 7 - 2x. This will allow us to plug 7 - 2x into the second equation for y. Like this:

If we simplify and solve for x, this is what we get:

If we plug x back into the first equation, we can solve that y = 3.
I find that substitution will often make the math a little more complicated and increase the steps when compared with the combination method. However, it can be helpful in certain circumstances. I would advise that you be comfortable with both methods.