GMAT Equations From Scratch
Equations on the GMAT are less about algebraic technique than about picking the right route quickly. The methods you learned at school work, and several of them are slower than what the exam rewards.
One rule underneath everything
An equation is a balanced scale. Two oranges on the left, two on the right. Add three to both sides and it stays balanced. Add three to one side only and it tips.
That is the whole of equation manipulation. Whatever you do, do it to both sides. Every error in solving an equation traces back to breaking this.
You can also flip an equation around whenever it helps. If 7 = x is awkward, write x = 7.
An equation is not an expression
This catches people out and costs marks that should be free.
If a question asks for the value of 2x + 3 when x = 2, you substitute and calculate: 4 + 3 = 7. You are not solving for anything. There is no other side to balance.
The trap is that the working looks like an equation, so people start manipulating both sides. Multiply 4 + 3 = 7 by two and you get 14, which is true and useless, because the question asked for 7.
A simple guard: write "answer =" at the top of your notes and keep everything equal to that. If you find yourself changing the left-hand side, you have drifted into solving something nobody asked about.
Linear equations: two steps
A linear equation has the unknown with no power, no root, and no absolute value. Just x.
Step one: get all the x terms on one side and all the numbers on the other, applying the same operation to both sides.
Step two: divide or multiply to leave a single x.
For 2x + 5 = 13: subtract 5 from both sides to get 2x = 8, then divide both sides by 2 to get x = 4.
You can sometimes spot the answer without working. On a real GMAT question the numbers are rarely that friendly, so build the habit of doing it properly.
Quadratics: use product and sum
Put the equation in standard form first: everything on one side, equal to zero, in the shape ax² + bx + c = 0. Signs matter enormously here.
Then use two facts about the solutions:
- Their product is c/a
- Their sum is −b/a
For x² − 5x + 6 = 0: the product is 6, the sum is 5. List the simple pairs that multiply to 6 (2 and 3, 1 and 6, and their negatives), then check which pair sums to 5. Only 2 and 3 work, so those are your solutions.
This beats the quadratic formula on the GMAT for two reasons. It uses arithmetic you can do in your head, and it exploits the fact that GMAT solutions are almost always small whole numbers. Start with round numbers and you will usually find the pair in seconds.
Two things to watch. The value of a is 1 in roughly 80% of GMAT quadratics, so a question where it is not is signalling difficulty. And the minus sign in −b/a is a reliable source of errors: if b is −5, the sum is +5, not −5.
Simultaneous equations: two methods, and one is usually better
Substitution always works. Isolate one unknown, then plug it into the other equation.
Counterintuitively, isolate the variable you don't want. If you need y, isolate x and substitute it out, which leaves an equation in y alone.
Combination is usually faster when the numbers cooperate. Add or subtract whole equations to eliminate a variable. With x + y = 5 and x − y = 1, adding them cancels the y terms immediately and gives 2x = 6.
When the coefficients do not line up, scale first. For 2x + 3y = 8 and 3x + 2y = 7, multiply the first by 3 and the second by 2 to make both x terms 6x, then subtract.
The rule of thumb: look at the numbers, and if you can see a combination that eliminates something, take it. If not, fall back to substitution. Substitution always works and it tends to produce fractions and take longer.