{"id":5622,"date":"2024-04-19T09:00:51","date_gmt":"2024-04-19T16:00:51","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=5622"},"modified":"2020-01-15T10:48:17","modified_gmt":"2020-01-15T18:48:17","slug":"gmat-systems-of-equations","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/gmat-systems-of-equations\/","title":{"rendered":"GMAT Tricks with Systems of Equations"},"content":{"rendered":"

Many GMAT test-takers vaguely remember a rule from high school, that it\u2019s possible to solve for two variables if and only if<\/strong> you\u2019re given two equations, and generally that it\u2019s possible to solve for n<\/em> variables if and only if<\/strong> you\u2019re given n<\/em> equations. Unfortunately, that rule isn\u2019t quite correct as written, and even the correct rule isn\u2019t always relevant.<\/p>\n

Applying the rule incorrectly causes quite a few errors on the quant section, particularly with Data Sufficiency questions<\/a>. Sometimes, two equations aren\u2019t enough to allow us to solve for two variables, or even for one. That is, sometimes, information that seems sufficient isn\u2019t in fact sufficient. <\/p>\n

The trick was that the rule above isn\u2019t correct as written. The correct rule is that a system of n distinct linear equation is sufficient to solve for n variables, but that sometimes the GMAT gives you systems of equivalent (not distinct) equations, or exponential or quadratic (not linear) equations, and that the rule doesn\u2019t apply to such systems.<\/p>\n

Today, we\u2019ll see that sometimes information that doesn\u2019t seem sufficient turns out to be, even when the equations in question are distinct and linear. How does that happen?<\/p>\n

GMAT Systems of Equations: Sample Problem<\/h2>\n

Take a minute or two to answer this problem:<\/p>\n

Andres bought exactly two sorts of donuts, old-fashioned donuts and jelly donuts. If each old-fashioned donut costs $0.75 and each jelly donut costs $1.20, how many jelly donuts did Andres buy?<\/p>\n

(1) Andres bought a total of eight donuts.
\n(2) Andres spent exactly $7.35 on donuts.<\/p>\n

(A)<\/strong> Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
\n(B)<\/strong> Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
\n(C)<\/strong> BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
\n(D)<\/strong> EACH statement ALONE is sufficient.
\n(E)<\/strong> Statements (1) and (2) TOGETHER are NOT sufficient.<\/p>\n

In this case, if we too quickly apply the rule described above, we\u2019ll probably choose (C) after a bit of algebra. It turns out, though, that (C) is not the correct answer.
\n <\/p>\n

Let’s Translate This Into GMAT Algebra<\/h2>\n

Let j=the number of jelly donuts Andres purchased and 1.2j=the amount that Andres spent on jelly donuts. <\/p>\n

Note that we want to solve for j.<\/p>\n

Let f=the number of old-fashioned donuts Andres purchased and 0.75f=the amount that Andres spent on old-fashioned donuts.<\/p>\n

The question stem doesn\u2019t give us an equation. (Well, it could give us 1.2j+0.75f=t, where t=the total dollars spent on donuts, but that equation isn\u2019t useful.) <\/p>\n

We can rewrite statement (1) as an equation:<\/p>\n

(1) j+f=8<\/p>\n

Obviously that doesn\u2019t allow us to solve for j. Eliminate answers (A) and (D).<\/p>\n

We can rewrite statement (2) as an equation:<\/p>\n

1.2j+0.75f=7.35<\/p>\n

This doesn\u2019t seem to be sufficient either. After all, solving for j yields a weird variable expression, j=6.125-0.833\u2026f. So we\u2019ll probably eliminate (B) as well.<\/p>\n

You could solve for j using both statements together but you don\u2019t really need to do the math, since they\u2019re obviously distinct linear equations. It\u2019s enough to note that they are sufficient together without actually figuring out that j=3.
\n <\/p>\n

What’s Wrong With That Approach?<\/h2>\n

The trouble with that approach is that (2) is in fact sufficient. Yes, if we look at Statement (2) merely as an algebraic equation and we ignore the story that gave rise to the equation, then we have an infinite number of solutions for the pair j and f. Let\u2019s make a little function table, assigning simple integer values to f and letting those determine corresponding values for j:<\/p>\n