ACT Math Algebra: linear equations and inequalities, systems, quadratics, polynomials, exponentials and the coordinate plane

What the Algebra area covers

Algebra, about 12 to 15 percent of the ACT Math test, is one of its largest areas and the backbone of a high score. It runs from solving linear equations to quadratics, factoring and the coordinate plane. This guide ties together the dot points: linear equations and inequalities, systems of equations, quadratic equations, polynomials and factoring, exponential and radical equations, and expressions and the coordinate plane.

Linear equations and inequalities

Solve a linear equation by simplifying each side, collecting the variable on one side, and isolating it; check by substituting back. For inequalities, use the same steps but flip the sign when multiplying or dividing by a negative. A cancelling variable gives no solution (false statement) or all reals (true statement).

Systems of equations

Solve two linear equations by substitution (replace a variable with an expression) or elimination (add or subtract to cancel a variable). The solution is the lines' intersection. Parallel lines (same slope, different intercept) give no solution; identical lines give infinitely many. Many word problems hide a two-variable system.

Quadratic equations

Set to $ax^{2} + bx + c = 0$, then factor, take square roots (no $bx$ term), or use the quadratic formula $x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$. The discriminant $b^{2} - 4ac$ counts real roots (positive: two, zero: one, negative: none). Remember both signs when taking a square root.

Polynomials and factoring

Expand with FOIL and distribution. Factor in order: greatest common factor first, then patterns (difference of squares $a^{2} - b^{2} = (a-b)(a+b)$, perfect-square trinomials), then the general quadratic. Simplify rational expressions by factoring and cancelling factors (never terms).

Exponential and radical equations

For exponentials, match bases and equate exponents (rewrite numbers as powers of a common base). For radicals, isolate and square, then check for extraneous solutions, because squaring can introduce false roots.

Expressions and the coordinate plane

Evaluate expressions with careful sign work; solve literal equations for a variable as if the others were constants. Slope is $\frac{y_2 - y_1}{x_2 - x_1}$; a line is $y = mx + b$ or $y - y_1 = m(x - x_1)$. Parallel slopes are equal; perpendicular slopes are negative reciprocals. The midpoint averages coordinates; the distance is $\sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}}$.

Check your knowledge

Try these, then read the solutions.

1. Solve $3x - 4 = 2x + 9$. [1 point]
2. Solve the system $x + y = 8$, $x - y = 2$. [2 points]
3. Solve $x^{2} - x - 12 = 0$. [2 points]
4. Factor completely $4x^{2} - 25$. [1 point]
5. Find the slope between $(1, 3)$ and $(5, 11)$. [1 point]