Math 35 min

Algebra

Solving and graphing linear, quadratic and exponential equations, inequalities and systems.

Algebra is the engine of the whole Math test. Get fast at solving equations, lines and quadratics, and you’ll feel it on almost every page.

What it tests

Algebra questions ask you to solve, graph and build equations: linear equations and inequalities, systems of equations, quadratics, exponential models, and simplifying expressions. You’ll also translate word problems into equations.

How it shows up

The Math test is 45 questions in 50 minutes — a little over a minute each. Every question has four answer choices, a calculator is allowed on the whole section, there is no penalty for guessing, and no formula sheet is given.
Algebra is one of the larger Preparing for Higher Math categories, and algebra skills show up inside many other questions too.
Common forms: “What is the value of x?”, “Which equation represents the line…?”, “For what value of k does the system have no solution?”, and word problems that need an equation.

Key rules & concepts

The content you actually need to know.

Linear equations

Distribute, combine like terms, then get the variable alone with inverse operations.
Clear fractions first by multiplying every term by the common denominator.
Distributing a negative flips every sign inside: −2(x − 5) = −2x + 10.

Lines and slope

Slope m = (y₂ − y₁) / (x₂ − x₁) = rise over run.
Slope-intercept form: y = mx + b (b is the y-intercept). Point-slope form: y − y₁ = m(x − x₁).
Standard form Ax + By = C has slope −A/B.
Parallel lines have equal slopes. Perpendicular slopes are negative reciprocals (2 and −1/2).

Inequalities

Solve like equations — but FLIP the inequality sign when you multiply or divide by a negative.
|x| < a means −a < x < a. |x| > a means x < −a or x > a.

Systems of equations

Substitution: solve one equation for a variable and plug it into the other.
Elimination: add or subtract the equations to cancel a variable.
No solution: same slope, different intercepts (parallel lines). Infinitely many: the same line. Exactly one: different slopes.

Quadratics

Factor and use the zero product property: (x − 5)(x + 3) = 0 → x = 5 or x = −3.
Quadratic formula for ax² + bx + c = 0: x = [−b ± √(b² − 4ac)] / (2a).
Discriminant b² − 4ac: positive → 2 real solutions, zero → 1, negative → none (2 complex).
Sum of solutions = −b/a; product = c/a.
Vertex form y = a(x − h)² + k has vertex (h, k). In standard form the vertex’s x-coordinate is −b/(2a).
Patterns: (a + b)² = a² + 2ab + b²; a² − b² = (a − b)(a + b).

Exponential models

Growth: y = a(1 + r)^t. Decay: y = a(1 − r)^t, where r is the rate as a decimal.
A 15% decrease each year means multiply by 0.85 each year — not subtract 15% of the original every year.

Expressions and word problems

“5 less than x” is x − 5 (order matters!). “3 more than twice y” is 2y + 3.
Simplify rational expressions by factoring and canceling common FACTORS, never terms: (x² − 9)/(x + 3) = x − 3.

Strategy & traps

Step by step

1Translate the words into an equation before doing any arithmetic.
2Solve algebraically — or backsolve: plug the answer choices in, starting with a middle value.
3For “which expression is equivalent,” pick a number for x and test the choices.
4Reread the question: are they asking for x, or for something like 2x + 1?

Answering the wrong quantity

You solve for x = 4 but the question asks for x² − 1. The wrong-quantity answer is always a choice.

Forgetting to flip the inequality

Dividing by a negative number reverses < and >.

Sign errors when distributing

−3(x − 4) is −3x + 12. Write the step out.

Board Buddy tip: Circle what the question actually asks for. Solving for x is often only step one.

Worked examples

Try each one first, then reveal the solution.

Example 1

If 3(x − 4) + 2x = 18, what is the value of x?

A1.2
B4.4
C6
D7.2

Example 2

Which equation describes the line in the standard (x, y) coordinate plane that passes through (2, 5) and (6, −3)?

Ay = −2x + 9
By = 2x + 1
Cy = −½x + 6
Dy = −2x + 1

Example 3

If 2x + y = 11 and x − y = 1, what is the value of xy?

Practice

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Question 1

Which of the following is the solution set of −3x + 7 > 22?

Question 2

What is the sum of the solutions of x² − 2x − 15 = 0?

Question 3

For what value of k does the system 4x − 2y = 7 and kx − 3y = 5 have no solution?

Question 4

A phone is worth $800 when new, and its value decreases by 15% each year. To the nearest cent, what is its value after 3 years?

Question 5

For all x ≠ −3 and x ≠ 2, which expression is equivalent to (x² − 9)/(x² + x − 6)?