How do you find the equation of a line on the ACT?

Find the slope with rise over run, m = (y2 - y1)/(x2 - x1), then substitute one known point into y = mx + b to solve for the intercept b. Write the finished equation in slope-intercept form.

When do you flip the inequality sign?

Flip the inequality sign only when you multiply or divide both sides by a negative number. Adding, subtracting, or dividing by a positive number never changes the direction of the inequality.

What is the slope of a line perpendicular to another line?

A perpendicular slope is the negative reciprocal of the original slope. Flip the fraction and change its sign so that the two slopes multiply to -1. For example, the perpendicular of 3/4 is -4/3.

How do I turn an ACT word problem into a linear equation?

Identify the fixed starting amount (a flat fee or initial value) as the intercept b, and the constant per-unit rate as the slope m. Then write total = m times the variable plus b.

CramClub • Linear Equations and Inequalities

$m = \dfrac{y_2 - y_1}{x_2 - x_1}$

$m_1 \cdot m_2 = -1$

Multiply or divide by a negative $\Rightarrow$ reverse the sign.

Linear Relationships Are Everywhere on the ACT

A linear equation describes a quantity that changes at a constant rate. Its graph is a straight line, and on the ACT Math section it is the single most common algebraic object you will meet — in pure-algebra problems, in coordinate geometry, and hidden inside word problems about cost, distance, and rate. Master the linear toolkit and you unlock a large slice of the 60-question test.

The defining feature of a line is a constant slope: for every one-unit increase in $x$ , the value of $y$ changes by the same fixed amount. That amount is the slope $m$ , and the point where the line crosses the $y$ -axis is the $y$ -intercept $b$ .

Slope-Intercept Form

The most useful way to write a line is slope-intercept form:

$y = mx + b$

Here $m$ is the slope and $b$ is the $y$ -intercept. If you know any two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line, the slope is the rise over run:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

For example, the line through $(1, 4)$ and $(3, 10)$ has slope $m = \frac{10 - 4}{3 - 1} = \frac{6}{2} = 3$ . Substituting one point into $y = 3x + b$ gives $4 = 3(1) + b$ , so $b = 1$ and the line is $y = 3x + 1$ .

Solving a Linear Equation

To solve a linear equation for one unknown, undo the operations in reverse order: clear parentheses, collect the variable on one side and constants on the other, then divide.

Solve $5(x - 2) + 3 = 2x + 10$ .

5x - 10 + 3 &= 2x + 10 \\ 5x - 7 &= 2x + 10 \\ 3x &= 17 \\ x &= \frac{17}{3} \end{aligned}$$ A clean, non-integer answer is completely normal on the ACT — do not assume you made an error just because the result is a fraction. ## Parallel and Perpendicular Lines The ACT loves to test how slopes relate. | Relationship | Slopes | |---|---| | **Parallel** lines | Equal slopes: $m_1 = m_2$ | | **Perpendicular** lines | Negative reciprocals: $m_1 \cdot m_2 = -1$ | A line perpendicular to $y = \tfrac{2}{3}x + 5$ has slope $-\tfrac{3}{2}$, because $\tfrac{2}{3}$ and $-\tfrac{3}{2}$ multiply to $-1$. ## Linear Inequalities An **inequality** replaces the equals sign with $<$, $>$, $\le$, or $\ge$ and describes a whole range of values. Solve it exactly like an equation, with **one crucial exception**: when you multiply or divide both sides by a **negative** number, you must **reverse** the inequality sign. **Solve $-4x + 9 \ge 25$.** $$\begin{aligned} -4x &\ge 16 \\ x &\le -4 \end{aligned}$$ The sign flipped from $\ge$ to $\le$ because we divided by $-4$. Note the trap: only multiplying or dividing by a negative flips the sign. Adding or subtracting a negative never does. Confirm the direction by testing a value from your range — $x = -5$ gives $-4(-5) + 9 = 29 \ge 25$, which is true. ## Translating Word Problems Half of ACT linear questions are dressed up as stories. Train yourself to spot the two numbers that matter: a **starting value** (the intercept) and a **rate of change** (the slope). > A technician charges a flat \$40 service fee plus \$75 per hour of labor. The total cost for $h$ hours is $C = 75h + 40$. Here $40$ is the intercept (the cost before any labor) and $75$ is the slope (the cost per additional hour). Signal phrases for inequalities matter too: *at least* and *no less than* mean $\ge$; *at most* and *no more than* mean $\le$. ## A Reliable Game Plan - **Two points given?** Compute slope with rise over run, then solve for $b$. - **Solve for a variable?** Distribute, collect, divide — a fraction answer is fine. - **Parallel or perpendicular?** Equal slopes, or negative reciprocals. - **Inequality?** Same steps as an equation, but flip the sign when you divide by a negative. - **Word problem?** Identify the fixed starting value and the constant rate, then build $y = mx + b$. ## Common Mistakes to Avoid - Forgetting to flip the inequality when dividing by a negative number. - Reading the slope as the intercept (or vice versa) in a word problem. - Using the reciprocal *without* the negative sign for a perpendicular slope. - Distributing to only the first term inside parentheses.

Solve $3x - 7 = 11$ for $x$ .

A line passes through $(2, 3)$ and $(6, 11)$ . Write its equation in slope-intercept form.

Solve $4(2x - 1) = 3(x + 4) + 6$ .

A plumber charges a $60 visit fee plus $45 for each hour of work. If a job costs $240 in total, how many hours did the plumber work?

Solve $7 - 2x > 3x - 8$ and describe the solution.

Line $\ell$ has equation $y = \tfrac{3}{4}x + 2$ . Find the equation of the line perpendicular to $\ell$ that passes through $(6, 1)$ .

Forgetting to flip the inequality sign after dividing by a negative. Solving $-3x < 12$ as $x < -4$ is wrong.

Whenever you divide or multiply by a negative, reverse the sign: $-3x < 12 \Rightarrow x > -4$ . Test a value to confirm.

Using the plain reciprocal for a perpendicular slope — turning slope $\tfrac{2}{3}$ into $\tfrac{3}{2}$ instead of $-\tfrac{3}{2}$ .

Perpendicular slopes are negative reciprocals: flip the fraction AND change the sign so the product is $-1$ .

Swapping slope and intercept in a word problem, e.g. treating the flat fee as the per-unit rate.

The one-time or starting amount is the intercept $b$ ; the per-unit or per-hour amount is the slope $m$ . Label them before writing the equation.

Distributing to only the first term, writing $4(2x - 1) = 8x - 1$ .

Multiply the outside factor by every term inside: $4(2x - 1) = 8x - 4$ .

If $2x + 5 = 17$ , what is the value of $x$ ?

What is the slope of the line that passes through the points $(-1, 2)$ and $(3, 14)$ ?

A car rental costs a flat $35 plus $0.20 per mile driven. Which expression gives the total cost, in dollars, for driving $m$ miles?

What is the solution to the inequality $-5x + 3 \le 18$ ?

Which line is perpendicular to $y = -\tfrac{2}{5}x + 1$ ?

If $\tfrac{1}{3}(6x - 9) = 2x - 3$ is true for all $x$ , and $4(x + a) = 4x + 20$ , what is the value of $a$ ?