ACT Math Lines: 11 Practice Problems with Step-by-Step Explanations
The "Lines" chapter is one of the most heavily tested topics on the ACT Math section, showing up in some form on nearly every test — usually four to six questions, and sometimes more when word problems that model real situations with linear equations are included. At its core, this chapter tests four related skills: finding the slope of a line from two points, writing the equation of a line using slope-intercept form (y = mx + b) or point-slope form, finding the midpoint or distance between two points in the coordinate plane, and translating linear models — equations like 23h + 75 = 236 — into their real-world meaning.
The formulas themselves are simple, but ACT questions are built to catch small errors. The most common mistake is a sign error in the slope formula, mixing up which point's x and y values come first, which flips the sign of the answer and lands exactly on a wrong answer choice. Another frequent trap is confusing perpendicular and parallel slopes: parallel lines share the exact same slope, while perpendicular lines have slopes that are negative reciprocals of one another. On word-problem-style questions, the ACT often asks you to interpret one specific term in an equation — the constant or a coefficient — rather than solve for a variable, so reading the equation in context matters just as much as the algebra.
Work through the 11 problems below — a mix of slope, midpoint, distance, equation-writing, and interpretation questions — then click to reveal the full step-by-step explanation for each one.
On this page:
- Problem 1 — Slope from a point and x-intercept
- Problem 2 — Midpoint of a segment
- Problem 3 — Slope from two points
- Problem 4 — Solving for a point on a line
- Problem 5 — Perpendicular slope
- Problem 6 — Interpreting a constant
- Problem 7 — Linear model, rate of change
- Problem 8 — Interpreting a coefficient
- Problem 9 — Y-intercept in terms of a variable
- Problem 10 — Writing the equation of a line
- Problem 11 — Distance formula
Practice Problems
In the standard (x, y) coordinate plane, a line intersects the x-axis at (−1, 0) and contains the point (7, 2). What is the slope of the line?
- A) 1/4
- B) 1/3
- C) 7/3
- D) 4
The x-intercept (−1, 0) is a point on the line, so you have two points to work with: (−1, 0) and (7, 2). Slope is rise over run: (y₂ − y₁)/(x₂ − x₁) = (2 − 0)/(7 − (−1)) = 2/8 = 1/4.
A common error here is mixing up the subtraction order or mishandling the negative sign in 7 − (−1), which turns the denominator into 8 instead of 6 and gives a wrong answer.
In the standard (x, y) coordinate plane, what is the midpoint of the line segment with endpoints (2, 6) and (6, 10)?
- A) (3, 6)
- B) (4, −4)
- C) (−4, −4)
- D) (4, 8)
The midpoint formula averages the x-coordinates and the y-coordinates separately: ((x₁ + x₂)/2, (y₁ + y₂)/2) = ((2 + 6)/2, (6 + 10)/2) = (8/2, 16/2) = (4, 8).
Answer choices B and C are traps for students who subtract instead of add the coordinates.
A line in the standard (x, y) coordinate plane passes through the points (−7, 5) and (1, −4). The slope of the line is:
- A) −9/8
- B) −1/6
- C) 1/6
- D) 9/8
Using slope = (y₂ − y₁)/(x₂ − x₁) with (−7, 5) and (1, −4): (−4 − 5)/(1 − (−7)) = −9/8.
Keep the order of points consistent in both the numerator and denominator — if you swap the order in one but not the other, you'll flip the sign and land on 9/8 instead.
The point (5, m) lies on the graph of the line y = −2/5x + 11 in the standard (x, y) coordinate plane. What is the value of m?
- A) 6
- B) 9
- C) 11
- D) 13
Since (5, m) is on the line, substitute x = 5 into the equation: m = −2/5(5) + 11 = −2 + 11 = 9.
The most common mistake is forgetting to multiply the fraction correctly or dropping the negative sign, which lands on answer choice C (11) instead.
If line A has the equation 12x = 11y + 145, what is the slope of a line perpendicular to line A?
- A) −11/12
- B) 11/12
- C) −11/145
- D) 145/12
First, rewrite line A in slope-intercept form. Starting from 12x = 11y + 145, solve for y: 11y = 12x − 145, so y = (12/11)x − 145/11. The slope of line A is 12/11.
A perpendicular line has a slope equal to the negative reciprocal of the original slope. Flip 12/11 to get 11/12, then negate it: −11/12.
To draw a mural, a painter charges a onetime fee and $23 per hour of work. The equation 23h + 75 = 236 represents this situation, where h is the number of hours worked. Which of the following is the best interpretation of the 75 in this context?
- A) The onetime fee, in dollars.
- B) The number of hours worked.
- C) The charge per hour, in dollars.
- D) The total charge, in dollars.
In this model, 23h is the variable cost — $23 for every hour worked — and 236 is the total charge on the right side of the equation. The 75 is the constant added regardless of how many hours are worked, which matches the onetime fee described in the problem.
23 (the coefficient of h) is the charge per hour, and 236 is the total charge — 75 is the only term left to represent the flat fee.
A doctor uses the model w = 9a + 19.5 to estimate the weight, w, of a boy in terms of the boy's age, a, in years. Based on the model, what is the expected increase in the boy's weight, in pounds, from his 3rd to 7th birthday?
- A) 9
- B) 18
- C) 27
- D) 36
In this linear model, 9 is the slope — the rate of change in weight per year. From age 3 to age 7 is a change of 4 years, so the expected weight increase is 9 × 4 = 36 pounds.
You don't need to calculate the actual weight at age 3 and age 7 and subtract — the slope alone tells you the rate of change, which is the fastest path to the answer.
A 4,864-piece jigsaw puzzle has 276 edge pieces, and the rest are pieces inside of the puzzle. The equation 74x + 276 = 4,864 describes this situation, where x represents the number of rows that contain inside pieces. Which of the following is the best interpretation of 74x in this context?
- A) There are 74x total pieces.
- B) There are 74x pieces in each row.
- C) There are 74x edge pieces.
- D) There are 74x inside pieces.
The total puzzle has 4,864 pieces, and 276 of them are edge pieces (a fixed constant, not multiplied by x). That means the remaining term, 74x, must represent everything that isn't an edge piece — the inside pieces. Since 74x + 276 = 4,864, isolating 74x gives 74x = 4,864 − 276 = 4,588, the total count of inside pieces.
74 by itself is the number of inside pieces per row, but the question asks about 74x — the full product — which represents the total inside pieces across all x rows.
A line in the xy-plane has a slope of 3 and passes through the point (2a, 6a), where a is a nonzero constant. What is the y-intercept of this line in terms of a?
- A) 0a
- B) 3a
- C) 16/3 a
- D) 12a
Use point-slope form: y − y₁ = m(x − x₁). Plugging in the slope 3 and the point (2a, 6a): y − 6a = 3(x − 2a).
Distribute and simplify: y = 3x − 6a + 6a = 3x + 0. The y-intercept — the constant term — is 0, which matches answer choice A (written as 0a to match the form of the other choices).
Line k passes through the points (−2, 10) and (4, 28). Which of the following is the equation for line k?
- A) y = 3x + 16
- B) y = 3x + 4
- C) y = 1/3x + 12
- D) y = 1/3x + 15
First find the slope: (28 − 10)/(4 − (−2)) = 18/6 = 3. Then use point-slope form with either point, for example (−2, 10): y − 10 = 3(x − (−2)) = 3(x + 2) = 3x + 6.
Add 10 to both sides: y = 3x + 16.
In the standard (x, y) coordinate plane, the point (1, 7) is 13 coordinate units away from which of the following points?
- A) (6, −5)
- B) (4, −8)
- C) (−7, −2)
- D) (−4, 18)
Use the distance formula: distance = √((x₂ − x₁)² + (y₂ − y₁)²), or square both sides to avoid the square root: (x₂ − x₁)² + (y₂ − y₁)² = 13² = 169.
Testing (6, −5): (6 − 1)² + (−5 − 7)² = 5² + (−12)² = 25 + 144 = 169. This matches, confirming a distance of exactly 13 units. This is a 5-12-13 Pythagorean triple in disguise — recognizing that pattern can save time over computing the square root directly.
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