Algebra: complete GMAT guide

GMAT algebra questions usually fall into one of five groups:

• Linear equations and systems of equations
• Quadratic equations and expressions
• Algebraic manipulation, including factoring and identities
• Word problems that translate into equations
• Properties of expressions, such as signs, ranges, and integer constraints

The arithmetic itself is rarely hard. The difficulty comes from deciding what to solve for, which variable to eliminate, or which expression can be rewritten. A question that looks like it needs several minutes of algebra often has a shortcut based on the structure of the expression.

A linear equation has variables raised only to the first power. The standard form is $ax+by=c$. A system of two linear equations can be solved by substitution or elimination. Elimination is usually faster when the coefficients are already close to opposites.

Substitution

Solve one equation for one variable, then replace that variable in the other equation.

Elimination

Multiply one or both equations so that adding them removes a variable.

When a system has no solution, the lines are parallel. When it has infinitely many solutions, the equations describe the same line. On the GMAT, a question may ask whether a system has a unique solution without asking you to find it. In that case, compare the ratio of the coefficients. For $a_1x+b_1y=c_1$ and $a_2x+b_2y=c_2$, a unique solution requires that $a_1/a_2 \neq b_1/b_2$.

A quadratic equation has the form $ax^2+bx+c=0$, with $a \neq 0$. The GMAT uses three main tools: factoring, the quadratic formula, and the discriminant.

Factoring

Look for two numbers whose product is $ac$ and whose sum is $b$. For $x^2-5x+6=0$, the numbers are $-2$ and $-3$, so the equation factors as $(x-2)(x-3)=0$. The solutions are $x=2$ and $x=3$.

The quadratic formula

When factoring is not obvious, use

The expression under the square root, $b^2-4ac$, is the discriminant. It tells you how many real solutions exist:

• Positive discriminant: two distinct real solutions
• Zero discriminant: one repeated real solution
• Negative discriminant: no real solutions

GMAT questions often use the discriminant to ask whether a quadratic has real roots, or to force a quadratic to have exactly one root. If a question says a quadratic has one real solution, set the discriminant equal to zero.

Vieta's relationships

For $ax^2+bx+c=0$, the sum of the roots is $-b/a$ and the product is $c/a$. These relationships save time when a question asks for a combination of roots rather than the roots themselves.

Several identities appear often enough that you should recognize them immediately:

• $(x+y)^2=x^2+2xy+y^2$
• $(x-y)^2=x^2-2xy+y^2$
• $x^2-y^2=(x-y)(x+y)$
• $(x+y)^3=x^3+3x^2y+3xy^2+y^3$
• $x^3+y^3=(x+y)(x^2-xy+y^2)$
• $x^3-y^3=(x-y)(x^2+xy+y^2)$

The difference of squares is the most useful identity on the GMAT. Any time you see something like $99^2-1$, rewrite it as $(99-1)(99+1)=98 \cdot 100=9800$ instead of squaring 99.

This pattern appears in several forms. If you need $x^3+\frac{1}{x^3}$, cube the original equation or multiply the squared result by the original expression. The goal is to build the target from the given expression, not to solve for $x$.

Most algebra word problems become easier when you assign variables in a sentence. Write down what each variable represents, then translate each fact into an equation. Avoid assigning a variable to a quantity you do not need.

For questions with ages, consecutive integers, or digits, define the variables carefully. If a two-digit number has tens digit $t$ and units digit $u$, the number is $10t+u$, not $tu$. If the digits are reversed, the new number is $10u+t$.

GMAT algebra often adds a constraint that changes the answer. The most common constraints are:

• The variables must be integers.
• The variables must be positive.
• The variables represent counts, so they cannot be negative or fractional.
• A quadratic has exactly one solution.
• An expression must be defined, so a denominator cannot be zero.

Check these constraints before you choose an answer. A question may produce two algebraic solutions, but only one of them is a positive integer. Another question may ask for the value of $x$ and include a denominator such as $x-2$, which means $x=2$ is not allowed.

Algebra is the most common content area in Data Sufficiency. Two ideas matter more than any formula.

First, one linear equation with two variables usually does not determine both variables. The equation $2x+3y=12$ has many solutions. But if the question asks for the value of $4x+6y$, one equation is enough, because $4x+6y=2(2x+3y)=24$. Always check whether the target is a multiple or combination of the given expression.

Second, a quadratic equation usually gives two possible values. If a Data Sufficiency statement yields $x=2$ or $x=-2$, the statement is not sufficient unless the question or the other statement eliminates one value. A statement that looks sufficient at first can fail because of a second solution.

1. Read the question and underline the target. Are you solving for a variable, a combination, or a yes/no condition?
2. Translate each sentence into an equation or inequality. Define every variable in words.
3. Look for structure before you calculate. Can you factor, substitute, or use an identity?
4. Choose the method that produces the target fastest: elimination, backsolving, plugging in numbers, or algebraic manipulation.
5. Apply constraints such as integer, positive, or nonzero.
6. Check the answer against the original question, not against an intermediate equation.

When the answer choices are numbers and the algebra looks messy, work backward from the choices. Start with the middle value, because the choices are usually ordered. If the result is too large, move down; if it is too small, move up. This method is especially effective on age problems, consecutive integer problems, and questions with percentages.

When the answer choices are expressions and the question has variables, plug in easy numbers. Use different numbers for different variables, avoid 0 and 1 unless the problem requires them, and check every choice. If two choices survive, plug in a second set of numbers. This method tests the expression's behavior without solving it.

• Solving for $x$ when the question asks for $2x+1$. Read the target twice.
• Forgetting that a quadratic can have two solutions. A statement that gives one root may not give the other.
• Dividing by a variable that could be zero.
• Assuming all variables are positive when the question does not say so.
• Using the wrong sign when moving terms across the equals sign.
• Treating $(x+y)^2$ as $x^2+y^2$.
• Missing a units conversion in a word problem.
• Choosing an answer that satisfies an intermediate equation but not the original condition.