Beat the System (of Equations)
Beat the System (of Equations)
So you’re kicking butt and taking names on those old one-variable equations, huh? Good. But some SAT questions contain two variables. Lucky for you, those questions also always contain two equations, and you can use the two equations in conjunction to solve the variables. These two equations together are called a system of equations. We said earlier that manipulating equations isn’t like manipulating your younger brother or sister. But solving systems of equations is like manipulating your younger brother and sister. You use one equation against the other, and in the end you get whatever you want.
There are two types of systems of equations that you’ll need to be able to solve for the SAT. The first, easier model involves substitution, and the second type involves manipulating equations simultaneously.
Substituting into the System
You know substitution: Find the value of one variable and then plug that into another equation to solve for a different variable.
If x – 4 = y – 3 and 2y = 6, what is x?
You’ve got two equations and you have to find x. The first equation contains both x and y. The second equation contains only y. To solve for x, you first have to solve for y in the second equation and substitute that value for y in the first equation. If 2y = 6, then y = 3. Now, substitute 3 in for y in x – 4 = y – 3:
Here’s one that’s more likely to give you trouble on the SAT:
Suppose 3x = y + 5 and 2y – 2 = 12k. Solve for x in terms of k.
In order to solve for x in terms of k, you have to first get x and k into the same equation. To make this happen, you can solve for y in terms of k in the second equation, and then substitute that value into the first equation to solve for x. (You could also solve this problem by solving for y in the first equation and substituting that expression in for y in the second equation.)
Then substitute y = 6k + 1 into the equation 3x = y + 5.
Solving Simultaneous Equations
Simultaneous equations are equations that both contain the same variables. You can use the equations to solve for the variables by using one of the equations to solve for one variable in terms of the other, and then substituting that expression into the other equation.
Suppose 2x + 3y = 5 and x + y = 7. What is x?
In this particular problem, you need to find x. But in order to find the value of x, you need to get that pesky y variable out of one of the equations, right? Here’s how to do it. First solve one of the equations for y in terms of x:
Now substitute 7 – x for y in the equation 2x + 3y = 5:
Here’s what just happened. You manipulated one equation to separate the two variables on either side of the equal sign. Then you substituted one side of that equal sign into the other equation so that only the variable whose value you had to find was left. Bold move!
Another Way to Solve Simultaneous Equations
So that’s how you can solve every simultaneous equation question on the SAT. But wait! There’s another, even faster way to solve simultaneous equations.
Some students find this method tricky, but it is definitely faster, and it works. The choice of which method to use is up to you. Take a look at the following question:
2x + 3y = 5 and –1x – 3y = –7. What is x?
The amazing thing about simultaneous equations is that you can actually add or subtract the entire equations from each other. Observe:
Here’s another example:
6x + 2y = 11 and 5x + y = 10. What is x + y?
By subtracting the second equation from the first,
In order to add or subtract simultaneous equations, you need to know what variable you want to solve for, and then add or subtract accordingly. But we’ve got to admit something: So far, we’ve purposely chosen very easy examples to show how this method works. You won’t always have two equations that you can immediately add or subtract from each other to isolate one variable:
2x + 3y = –6 and –4x + 16y = 13. What is the value of y?
You’re asked to solve for y, which means you’ve got to get rid of x. But how can you get rid of x if one equation has 2x and the other has –4x? Well, you can’t. But remember, you can change the form of the equation without changing the actual equation, as long as you do the same thing to both sides of the equation. For instance, you could multiply both sides of 2x + 3y = –6 by 2, which would give you
You can add this equation to –4x + 16y = 13 to isolate y.
On the SAT, you will almost always be able to manipulate one of the two equations in a pair of simultaneous equations so that they can be added and subtracted to isolate the variable you want. The question is whether you can see how to do it. Our recommendation? Since it’s faster, it always pays to take a second to try to see how to isolate the variable by adding or subtracting the equations. If you can’t, then go ahead and solve the simultaneous equations using the first method we described.
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