If you have two equations, you can solve for two variables.
This rule is a cornerstone of algebra. It’s how we solve for values when we’re given a relationship between two unknowns:
If I can buy 2 kumquats and 3 rutabagas for $16, and 3 kumquats and 1 rutabaga for $9, how much does 1 kumquat cost?
We set up two equations:
2k + 4r = 16
3k + r = 9
Then we can use either substitution or elimination to solve. (Try it out yourself; answer* below).
On the GMAT, you’ll be using the “2 equations à 2 variables” rule to solve for a lot of word problems like the one above, especially in Problem Solving. Be careful, though! On the GMAT this rule doesn’t always apply, especially in Data Sufficiency. Here are some sneaky exceptions to the rule…
2 Equations aren’t always 2 equations
On DS questions, the GMAT wants you to assume that if you have 2 equations, you can always solve for the values of 2 variables. Consider:
What is the value of x?
(1) 2x – y = 5x – 4
(2) 6x + y = 8 – y
At a first glance, we see that each statement has an equation with x and y in it. We assume that those aren’t sufficient on their own, but if we combine them, then 2 equations should allow us to solve for 2 variables. Right? Give it a try – see what values you get…
If you actually try to solve, you’ll see that statement (1) simplifies to 3x + y = 4. And statements (2) simplifies to… exactly the same thing! We didn’t really have 2 equations – we actually had 2 versions of the same equation. The answer would be E, not C.
2 variables aren’t always 2 variables
Try this one:
What is the value of m?
(1) m – 2n = 6
(2) 3m – n = 9 – (m + n)
Again, at a first glance it looks like each equation has 2 variables, so we’ll need both statements to solve. What happens when you simplify statement (2), though? The –n on each side will cancel, leaving us with a value for m. That’s sufficient!
We have to actually do the work to ensure that a) neither variable cancels out or b) we don’t secretly have the same equation. We can’t just jump to a conclusion without doing the work!
The “Combo”
If p + 3q = 6r, what is the value of p?
(1) 2r – q = 5
(2) r + 2q = 20
Here, if we combine the two statements, we can solve and find that q = 7 and r = 6, and then we can easily solve for the value of p. So it seems like the answer should be C, right? But of course the GMAT is sneakier than that…
Sometimes 2 equations will let us solve for the values of 2 variables, but that’s not what the question really asked! First, rephrase the question:
If p + 3q = 6r, what is the value of p?
p = 6r – 3q → what is the value of 6r – 3q?
→ what is 3(2r – q)?
→ what is 2r – q?
If that’s our question, then clearly statement (1) gives us a value for that expression. We didn’t need the value of each variable individually, we just needed a value for the “combo” of 2r – q.
The Integer Constraint
If I asked you to solve for x, and just gave you the equation 13x + 5y = 90, you wouldn’t be able to do it. One equation will never let you solve for two variables – that’s what we learned in high school. But see what happens when we have a word problem:
If xylophones cost $13 apiece and…. [I can’t think of an instrument that starts with ‘y.’ Let’s just say ‘zithers’] zithers cost $5 apiece at the Discount Music Emporium. If Wolfgang purchased at least one xylophone and one zither, then how many xylophones did Wolfgang buy?
(1) He spent a total of $90 on xylophones and zithers.
(2) He bought the same number of xylophones as zithers.
If we use both statements together, we get 13x + 5z = 90 and x = z. Two equations, so we can solve for two variables. But look again at the statements individually!
The second statement doesn’t help, because that “same number” could be anything. On the first statement, though, ask yourself: are there multiple combinations of $13 and $5 amounts that could add to $90?
Because the numbers of xylophones and zithers have to be positive integers, we’re just looking for combinations of multiples of 13 and multiples of 5. As it turns out, there’s only one combination that adds up to $90: 13(5) + 5(5). Statement (1) – just one single equation! – was sufficient because the integer constraint restricted it to one possibility.
The Quadratic
Another way that the GMAT will try to mess with our expectations about “2 equations → 2 variables” is by giving us two equations that create a quadratic:
What is the value of k?
(1) j + k = 9
(2) jk = 20
If we combine these two equations, we’ll get a quadratic:
20/k + k = 9
20 + k2 -9k = 0
(k – 5)(k – 4) = 0
We find that k could equal 4 or 5, but we’ll never know which. Quadratics give us two potential values, but they’re insufficient to give us a single value for a variable (unless there’s a constraint, or a perfect square). Here, the answer would be E.
Check your assumptions!
Remember, the GMAT likes to thwart your expectations! Don’t assume that two equations will always be needed (or sufficient) to solve for two variables.
For more examples of these “2 equations ≠ 2 variables,” try these problems. See if you can pinpoint which exception is being used in each:
OG 13/2015: DS #17, 23, 56, 59, 67, 68, 78, 114, 132, 156
* k = 2
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