Evaluate and rearrange algebraic expressions, solve literal equations for a variable, and find the slope and equation of a line in the coordinate plane (Algebra).

What this topic is asking

This topic covers two staples of ACT Algebra: manipulating expressions (evaluating, rearranging, solving for a variable in a formula) and the coordinate plane (slope and the equation of a line). Both are high-frequency, and both reward careful sign work.

Evaluating and rearranging expressions

Substitution is simple arithmetic, but sign care is everything.

Solving for a variable is the same inverse-operation process as solving a numeric equation; the only difference is the answer is an expression.

Slope in the coordinate plane

The slope measures steepness as rise over run.

The slope between two points is $m = \dfrac{y_2 - y_1}{x_2 - x_1}$. A positive slope rises left to right, a negative slope falls, a zero slope is horizontal, and an undefined slope (division by zero in the denominator) is vertical. Keep the two points in the same order in the numerator and denominator, or you will get the wrong sign.

The equation of a line

Two forms cover almost every ACT line question.

• Slope-intercept: $y = mx + b$, where $m$ is the slope and $b$ the $y$-intercept. Best when you know the slope and intercept.
• Point-slope: $y - y_1 = m(x - x_1)$, best when you know the slope and one point. Expand to slope-intercept if the answer choices use that form.

To find a line through two points, compute the slope, then use point-slope with either point.

Parallel and perpendicular lines

Slope relationships are a favourite ACT topic:

• Parallel lines have equal slopes.
• Perpendicular lines have slopes that are negative reciprocals: $m_1 m_2 = -1$, so a line perpendicular to one with slope $\frac{2}{3}$ has slope $-\frac{3}{2}$.

So a line through $(0, 1)$ perpendicular to $y = 2x + 5$ has slope $-\frac{1}{2}$ and equation $y = -\frac{1}{2}x + 1$.

Midpoint and distance

Two more coordinate formulas appear regularly. The midpoint of the segment from $(x_1, y_1)$ to $(x_2, y_2)$ is the average of the coordinates, $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$. The distance between them is $\sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}}$, which is the Pythagorean theorem on the horizontal and vertical gaps. These connect algebra to geometry and recur in the coordinate-geometry topic.

Why sign care wins points

Most errors here are not conceptual but arithmetic: a dropped negative when substituting, an inverted slope ratio, or a reciprocal taken without the sign change for a perpendicular line. Substituting values in parentheses, keeping point order consistent in the slope formula, and writing the negative reciprocal deliberately are the habits that turn near-misses into correct answers.

Try this

Q1. Evaluate $2x^{2} - 3y$ when $x = -3$ and $y = 4$. [1 point]

• Cue. $2(-3)^{2} - 3(4) = 18 - 12 = 6$.

Q2. Find the slope of the line through $(2, -1)$ and $(6, 7)$. [1 point]

• Cue. $\frac{7 - (-1)}{6 - 2} = \frac{8}{4} = 2$.