Data Sufficiency: algebra

Data Sufficiency: algebra | GMAT Sprint

Data Sufficiency algebra questions do not ask you to calculate a final value. They ask whether the information is enough to determine one answer. That shift changes the skill being tested: you need to recognize when an equation determines a target, when a quadratic leaves two possibilities, and when a constraint such as "positive integer" closes the gap.

How Data Sufficiency works

Each question has a question stem and two statements. You decide whether statement (1) alone is sufficient, whether statement (2) alone is sufficient, whether both together are sufficient, or whether even together they are not sufficient. The answer choices are fixed:

(A) Statement (1) alone is sufficient, but (2) alone is not.
(B) Statement (2) alone is sufficient, but (1) alone is not.
(C) Both statements together are sufficient, but neither alone is.
(D) Each statement alone is sufficient.
(E) The statements together are not sufficient.

Do not carry information from one statement into the other when testing them separately. Each statement is evaluated on its own first.

Start with the target

Before reading the statements, rewrite the question in your own words. Are you looking for a value, a relationship, or a yes/no answer? If the question asks for $x$ , then a statement that gives $x^2=9$ is not sufficient, because $x$ could be 3 or -3. If the question asks for $|x|$ , the same statement is sufficient, because both values give the same absolute value.

Sufficiency is about the target, not about how much information a statement contains. A statement can give several facts and still fail if the target remains ambiguous. A single equation can be sufficient if it determines the exact combination the question asks for.

One equation, two variables

A single linear equation with two variables usually does not determine both variables. The equation $3x+2y=12$ has many integer and non-integer solutions. But the same equation can determine a combination. If the question asks for $6x+4y$ , then $6x+4y=2(3x+2y)=24$ , so the statement is sufficient.

Example. What is the value of $x+y$ ?

(1) $2x+2y=10$

(2) $x-y=4$

Statement (1) divides to $x+y=5$ , so it is sufficient. Statement (2) gives a relationship but not the sum, so it is not sufficient. The answer is (A).

This pattern is common. The GMAT rewards looking at the target as an expression rather than as a pair of individual variables.

Quadratics and multiple solutions

A quadratic equation usually gives two solutions. A statement such as $x^2-5x+6=0$ means $x=2$ or $x=3$ . If the question asks for $x$ , the statement is not sufficient unless another condition eliminates one value. If the question asks for the product of the roots, the statement is sufficient because the product is 6 regardless of which root is which.

Example. What is the value of $x$ ?

(1) $x^2=16$

(2) $x \gt 0$

Statement (1) gives $x=4$ or $x=-4$ , so it is not sufficient. Statement (2) alone does not give a value. Together, $x=4$ , so the answer is (C).

Watch for the same issue with absolute value. $|x|=5$ gives two values. A condition such as $x \lt 0$ is needed to determine one.

Inequalities and yes/no questions

An inequality statement is sufficient for a yes/no question only if it forces the same answer for every allowed value. If the question asks "Is $x \gt 0$ ?" and a statement says $x^2 \gt 4$ , then $x$ could be 3 or -3. The statement does not answer the question consistently, so it is not sufficient.

Test the boundary cases: positive, negative, zero, and fractions between 0 and 1. A statement that works for positive integers may fail for fractions or negatives.

Example. Is $x \gt 1$ ?

(1) $x^2 \gt 1$

(2) $x \gt 0$

Statement (1) allows $x=2$ and $x=-2$ , so it is not sufficient. Statement (2) allows $x=0.5$ , so it is not sufficient. Together, $x \gt 0$ and $x^2 \gt 1$ give $x \gt 1$ . The answer is (C).

Integer constraints

An integer constraint can turn an insufficient equation into a sufficient one. The equation $2x+3y=12$ has many real solutions, but if $x$ and $y$ are positive integers, the possibilities are limited. Test small values to see whether only one pair works.

Be careful with the exact wording. "Positive integer" excludes zero. "Nonnegative integer" includes zero. "Integer" includes negatives. The allowed set is part of the statement, and it can decide sufficiency.

Systems of equations

Two distinct linear equations with two variables usually determine both variables, but not always. If the equations are multiples of each other, they represent the same line and do not add new information. For example, $x+y=4$ and $2x+2y=8$ are the same equation. Together they do not determine $x$ and $y$ .

If the question asks for a combination, two equations may be more than enough. If it asks for individual values, check whether the equations are independent.

Using algebra to test sufficiency

When testing a statement, translate it into an equation and compare it with the target. If the target is a combination such as $x+y$ , a statement that gives $2x+2y=12$ is sufficient because it determines $x+y=6$ . A statement that gives $x-y=2$ is not sufficient, because it does not determine the sum.

For systems, count independent equations and check that they are not multiples of each other. Two equations can look different and still be dependent. If one is a multiple of the other, they provide only one relationship.

Testing values efficiently

Use values that are easy to compute but still challenge the statement. If the statement involves a square, test a positive and a negative value. If it involves a fraction, test a value between 0 and 1. If it involves a denominator, test the value that makes the denominator zero to see whether it is excluded.

Example. Is $x \gt 1$ ?

(1) $x^3 \gt 1$

For real numbers, $x^3 \gt 1$ implies $x \gt 1$ because the cube function is increasing. Statement (1) is sufficient. Testing $x=2$ confirms the yes case, and testing $x=0.5$ shows that it fails the statement. The answer is (A).

Algebra can confirm what testing suggests. If the statement can be manipulated into the target without extra assumptions, it is sufficient. If the manipulation requires dividing by a variable of unknown sign, the statement may not be sufficient.

A repeatable approach

Rewrite the question and identify the target.
Evaluate statement (1) alone. Test several allowed values.
If (1) is sufficient, the answer is (A) or (D).
If (1) is not sufficient, evaluate statement (2) alone.
If (2) is sufficient, the answer is (B).
If neither is sufficient alone, combine them and check again.
Do not use information from one statement while testing the other.

Common traps

Solving for $x$ when the question asks for $x+y$ .
Assuming a quadratic has one solution.
Forgetting that a statement can be sufficient without giving individual values.
Carrying information between statements.
Ignoring negative or fractional values.
Treating dependent equations as independent.
Spending time computing a value when a yes/no answer is enough.

Quick reference

Situation	Usually sufficient?
One linear equation, target is the same combination	Yes
One linear equation, target is one variable	No
Quadratic, target is one variable	No, unless a constraint removes a root
Two independent linear equations	Yes for both variables
Two dependent linear equations	No
Inequality, yes/no target	Only if every allowed value gives the same answer

Data Sufficiency algebra is about ambiguity, not calculation. Ask whether the target can take more than one value. If it can, the statement is not sufficient. If every allowed value gives the same answer, it is.