ACT Math: Lines
Lines is a core Coordinate Geometry topic on the ACT, and slope is the single idea that ties nearly all of it together. This free, complete lesson covers the slope formula and types of slope, the three main equation forms of a line (slope-intercept, point-slope, and standard form), parallel and perpendicular line relationships, and intercepts and writing equations from points or graphs. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, in the same four-answer-choice, calculator-allowed format as the current ACT. Everything here is free, and you can keep practicing afterward with the full ACT Math Question Bank linked below.
1. The Slope Formula and Types of Slope
The slope of a line measures its steepness, "rise over run." Given two points (x₁, y₁) and (x₂, y₂):
m = y₂ − y₁x₂ − x₁
| Line type | Slope |
|---|---|
| Rises left to right | Positive |
| Falls left to right | Negative |
| Horizontal | 0 |
| Vertical | Undefined |
Worked Example: Find the slope of the line through (5, −2) and (6, 7).
m = 7 − (−2)6 − 5 = 91 = 9
1. Find the slope of the line through (2, 3) and (6, 11).
Explanation: m = 11 − 36 − 2 = 84 = 2.
2. Find the slope of the line through (−1, 4) and (3, −8).
Explanation: m = −8 − 43 − (−1) = −124 = −3.
3. What is the slope of a horizontal line?
Explanation: A horizontal line has no vertical rise, so its slope is 0.
4. What is the slope of a vertical line?
Explanation: A vertical line has no horizontal run, and division by zero is undefined, so its slope is undefined.
5. Find the slope of the line through (4, 7) and (4, −3).
Explanation: Both points share x = 4, making this a vertical line, so the slope is undefined.
2. Equations of Lines
| Form | Equation | Best used when... |
|---|---|---|
| Slope-intercept | y = mx + b | You know the slope and y-intercept |
| Point-slope | y − y₁ = m(x − x₁) | You know the slope and one point |
| Standard | Ax + By = C | Finding intercepts quickly |
Worked Example: Write the equation of a line with slope 3 passing through (2, 5), in slope-intercept form.
y − 5 = 3(x − 2)
y = 3x − 6 + 5 = 3x − 1
1. Write the equation of a line with slope 4 passing through (1, 6), in slope-intercept form.
Explanation: y − 6 = 4(x − 1), so y = 4x − 4 + 6 = 4x + 2.
2. Write the equation of a line passing through (0, −3) with slope −2, in slope-intercept form.
Explanation: Since the point (0, −3) has x = 0, −3 is already the y-intercept: y = −2x − 3.
3. Convert the equation 3x + 2y = 12 to slope-intercept form.
Explanation: 2y = −3x + 12, so y = −32x + 6.
4. Write the equation of the line passing through (2, −1) and (5, 8), in slope-intercept form.
Explanation: Slope: 8 − (−1)5 − 2 = 93 = 3. Using point-slope with (2, −1): y + 1 = 3(x − 2), so y = 3x − 6 − 1 = 3x − 7.
5. What is the slope and y-intercept of the line y = −5x + 9?
Explanation: In y = mx + b, m = −5 is the slope, and b = 9 is the y-intercept.
3. Parallel and Perpendicular Lines
Parallel lines have the exact same slope. Perpendicular lines have slopes that are opposite reciprocals of each other, their product is always −1.
Worked Example: Find the slope of a line parallel to y = 4x − 3.
Parallel lines share the same slope: 4.
1. A line is parallel to y = 2x + 7. What is its slope?
Explanation: Parallel lines have the same slope: 2.
2. A line is perpendicular to y = 2x + 7. What is its slope?
Explanation: The perpendicular slope is the opposite reciprocal of 2, which is −12.
3. Write the equation of a line parallel to y = 3x − 4 that passes through (2, 5).
Explanation: Parallel means slope = 3. y − 5 = 3(x − 2), so y = 3x − 6 + 5 = 3x − 1.
4. Write the equation of a line perpendicular to y = −12x + 6 that passes through (4, 3).
Explanation: The perpendicular slope of −12 is 2. y − 3 = 2(x − 4), so y = 2x − 8 + 3 = 2x − 5.
5. Two lines have slopes 34 and −43. Are these lines parallel, perpendicular, or neither?
Explanation: Their product is 34 × (−43) = −1, confirming the lines are perpendicular.
4. Intercepts and Writing Equations from Points
The x-intercept is where a line crosses the x-axis (set y = 0 and solve). The y-intercept is where it crosses the y-axis (set x = 0 and solve). Given two points, including two intercepts, you can find the slope and then write the full equation.
Worked Example: Find the x- and y-intercepts of 2x + 3y = 12.
x-intercept (set y = 0): 2x = 12, so x = 6, giving (6, 0).
y-intercept (set x = 0): 3y = 12, so y = 4, giving (0, 4).
1. Find the x-intercept of the line 4x − 5y = 20.
Explanation: Set y = 0: 4x = 20, so x = 5.
2. Find the y-intercept of the line 4x − 5y = 20.
Explanation: Set x = 0: −5y = 20, so y = −4.
3. A line has x-intercept 6 and y-intercept −3. What is its slope?
Explanation: The intercepts are the points (6, 0) and (0, −3). Slope: 0 − (−3)6 − 0 = 36 = 12.
4. A line passes through the points (0, 5) and (10, 0). What is its equation in slope-intercept form?
Explanation: Slope: 0 − 510 − 0 = −12. Since (0, 5) is already the y-intercept: y = −12x + 5.
Common Mistakes to Avoid
- Subtracting the coordinates in the wrong order. The slope formula requires subtracting y-values and x-values in the same order (both point 2 minus point 1, or both point 1 minus point 2), mixing the order flips the sign.
- Mixing up which line type has slope 0 versus an undefined slope. A horizontal line has slope 0 (flat, no rise). A vertical line has undefined slope (infinite steepness, division by zero).
- Using the same slope for a perpendicular line. A perpendicular line needs the opposite reciprocal of the original slope, not the same slope (that would make it parallel instead).
- Forgetting to flip the sign when finding a reciprocal. The opposite reciprocal of 23 is −32, not 32, both the flip and the sign change matter.
- Forgetting to set the other variable to 0 when finding an intercept. Finding the x-intercept means setting y = 0 (not x = 0), and finding the y-intercept means setting x = 0, mixing these up gives the wrong point entirely.
Frequently Asked Questions
Which form of a line's equation should I use?
Use slope-intercept form (y = mx + b) when you want to read off the slope and y-intercept directly, or when graphing. Use point-slope form (y − y₁ = m(x − x₁)) when you're given a slope and a point but not the y-intercept. Use standard form (Ax + By = C) when you need to quickly find both intercepts.
How can I quickly tell if two lines are parallel or perpendicular from their equations?
Rewrite both equations in slope-intercept form and compare the coefficients of x. If the slopes are identical, the lines are parallel. If multiplying the two slopes together gives exactly −1, they're perpendicular.
Why does dividing by zero make a vertical line's slope undefined, instead of just very large?
The slope formula divides the change in y by the change in x. For a vertical line, every point shares the same x-value, so the change in x is exactly 0, and division by 0 has no defined numerical result, no matter how the change in y behaves.
Where can I practice more problems like these?
The Lines quizzes in the ACT Math Question Bank include additional original problems on this topic, along with quizzes covering every other topic tested on the ACT Math section.