Linear equations and systems

Why linear algebra is worth fast hands

GMAT Quantitative Reasoning is 21 problem-solving questions in 45 minutes, and you cannot use a calculator in that section. Linear equations appear on their own and inside nearly every word problem, so the gain is not in knowing how to solve them (you do) but in solving them in a few lines, with no sign slips, and in noticing when the question lets you skip most of the work.

One variable: undo in reverse order

An equation stays true if you do the same thing to both sides. To isolate $x$, undo what was done to it in reverse order: first remove what was added, then what was multiplied.

$$\frac{3x - 4}{5} = 7 \;\Rightarrow\; 3x - 4 = 35 \;\Rightarrow\; 3x = 39 \;\Rightarrow\; x = 13$$

Clear fractions early by multiplying both sides by a common denominator, and expand brackets before collecting terms. When the variable appears on both sides, gather it on the side where its coefficient stays positive; fewer negative signs means fewer slips.

Two variables: elimination or substitution

A single equation in two unknowns, such as $2x + 3y = 12$, has infinitely many solutions: every point of a line. A second, independent equation pins down one point.

• Elimination. Scale one or both equations so that one variable has opposite coefficients, then add. It is usually fastest when coefficients are already matched or are small multiples of each other.
• Substitution. Solve one equation for a variable and put the expression into the other. It is fastest when one variable already has coefficient 1, as in $x - y = 6$.

Whichever you use, check the pair in the equation you did not use last. That costs ten seconds and catches most arithmetic errors.

Aim at the quantity asked for

Many questions ask for a combination, not for $x$ and $y$ separately. Before solving, look at what adding or subtracting the equations produces.

If $4a + 7b = 30$ and $7a + 4b = 36$, adding gives $11a + 11b = 66$, so $a + b = 6$ in one line. Subtracting the first from the second gives $3a - 3b = 6$, so $a - b = 2$. If the question wanted $a + b$, you are already done; solving fully would have taken three times as long.

The habit is simple: read the last line of the question first, then ask whether some sum, difference or multiple of the given equations is that quantity, or a multiple of it.

How many solutions?

Two linear equations in two unknowns behave like two lines:

Situation	Lines	Solutions
Coefficients not in proportion	cross once	exactly one
Coefficients in proportion, constants not	parallel	none
Everything in proportion	the same line	infinitely many

So $3x - 4y = 5$ and $6x - 8y = 7$ have no solution: the left sides are in ratio $1:2$, but the right sides are not. And $x + 2y = 4$ with $3x + 6y = 12$ is really one equation written twice. Two equations do not always mean one answer; that idea returns in data sufficiency.

Words into equations

Most linear questions arrive as a story. Name each unknown with a letter that reminds you what it counts ($a$ for adult tickets, $c$ for child tickets), write one equation per fact, and check the units: a count equation adds counts, a money equation adds price times count. When the story has two kinds of thing, a total number and a total value, it is a two-variable system; substituting the count equation into the value equation is usually quickest.