Lines | Free ACT Math Course | The School of Mathematics

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<h1>ACT Math: Lines</h1>

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.

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/act-lines-quiz-1">Practice Lines Free</a>

<a class="cta-btn cta-secondary" href="https://theschoolofmathematics.com/quiz/course/ACT-Math-Qbank">Explore the Full ACT Math Qbank</a>

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<h2>What's covered in this lesson</h2>

<ol>

<li><a href="#slope">The Slope Formula and Types of Slope</a></li>

<li><a href="#equations">Equations of Lines</a></li>

<li><a href="#parallel-perpendicular">Parallel and Perpendicular Lines</a></li>

<li><a href="#intercepts">Intercepts and Writing Equations from Points</a></li>

<li><a href="#mistakes">Common Mistakes to Avoid</a></li>

<li><a href="#faq">Frequently Asked Questions</a></li>

</ol>

</nav>

<h2 id="slope">1. The Slope Formula and Types of Slope</h2>

<p>The <strong>slope</strong> of a line measures its steepness, "rise over run." Given two points (x₁, y₁) and (x₂, y₂):</p>

<tr><th>Line type</th><th>Slope</th></tr>

<tr><td>Rises left to right</td><td>Positive</td></tr>

<tr><td>Falls left to right</td><td>Negative</td></tr>

<tr><td>Horizontal</td><td>0</td></tr>

<tr><td>Vertical</td><td>Undefined</td></tr>

</table>

<p><strong>Worked Example:</strong> Find the slope of the line through (5, −2) and (6, 7).</p>

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<p class="prompt">1. Find the slope of the line through (2, 3) and (6, 11).</p>

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<p><span class="label">Explanation:</span> m = <span class="frac"><span class="num">11 − 3</span><span class="den">6 − 2</span></span> = <span class="frac"><span class="num">8</span><span class="den">4</span></span> = 2.</p>

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<p class="prompt">2. Find the slope of the line through (−1, 4) and (3, −8).</p>

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<p><span class="label">Explanation:</span> m = <span class="frac"><span class="num">−8 − 4</span><span class="den">3 − (−1)</span></span> = <span class="frac"><span class="num">−12</span><span class="den">4</span></span> = −3.</p>

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<p class="prompt">3. What is the slope of a horizontal line?</p>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-3',false)">C) Undefined</button>

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<p><span class="label">Explanation:</span> A horizontal line has no vertical rise, so its slope is 0.</p>

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<p class="prompt">4. What is the slope of a vertical line?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pa-4',true)">A) Undefined</button>

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<p><span class="label">Explanation:</span> A vertical line has no horizontal run, and division by zero is undefined, so its slope is undefined.</p>

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<p class="prompt">5. Find the slope of the line through (4, 7) and (4, −3).</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pa-5',true)">A) Undefined</button>

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<p><span class="label">Explanation:</span> Both points share x = 4, making this a vertical line, so the slope is undefined.</p>

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<h2 id="equations">2. Equations of Lines</h2>

<tr><th>Form</th><th>Equation</th><th>Best used when...</th></tr>

<tr><td>Slope-intercept</td><td>y = mx + b</td><td>You know the slope and y-intercept</td></tr>

<tr><td>Point-slope</td><td>y − y₁ = m(x − x₁)</td><td>You know the slope and one point</td></tr>

<tr><td>Standard</td><td>Ax + By = C</td><td>Finding intercepts quickly</td></tr>

</table>

<p><strong>Worked Example:</strong> Write the equation of a line with slope 3 passing through (2, 5), in slope-intercept form.</p>

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<p class="prompt">1. Write the equation of a line with slope 4 passing through (1, 6), in slope-intercept form.</p>

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<p><span class="label">Explanation:</span> y − 6 = 4(x − 1), so y = 4x − 4 + 6 = 4x + 2.</p>

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<p class="prompt">2. Write the equation of a line passing through (0, −3) with slope −2, in slope-intercept form.</p>

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<p><span class="label">Explanation:</span> Since the point (0, −3) has x = 0, −3 is already the y-intercept: y = −2x − 3.</p>

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<p class="prompt">3. Convert the equation 3x + 2y = 12 to slope-intercept form.</p>

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<p><span class="label">Explanation:</span> 2y = −3x + 12, so y = −<span class="frac"><span class="num">3</span><span class="den">2</span></span>x + 6.</p>

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<p class="prompt">4. Write the equation of the line passing through (2, −1) and (5, 8), in slope-intercept form.</p>

</div>

<p><span class="label">Explanation:</span> Slope: <span class="frac"><span class="num">8 − (−1)</span><span class="den">5 − 2</span></span> = <span class="frac"><span class="num">9</span><span class="den">3</span></span> = 3. Using point-slope with (2, −1): y + 1 = 3(x − 2), so y = 3x − 6 − 1 = 3x − 7.</p>

</div>

<p class="prompt">5. What is the slope and y-intercept of the line y = −5x + 9?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pb-5',true)">A) Slope = −5, y-intercept = 9</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-5',false)">B) Slope = 9, y-intercept = −5</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-5',false)">C) Slope = 5, y-intercept = 9</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-5',false)">D) Slope = −5, y-intercept = −9</button>

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<p><span class="label">Explanation:</span> In y = mx + b, m = −5 is the slope, and b = 9 is the y-intercept.</p>

</div>

<h2 id="parallel-perpendicular">3. Parallel and Perpendicular Lines</h2>

<p><strong>Parallel</strong> lines have the exact same slope. <strong>Perpendicular</strong> lines have slopes that are opposite reciprocals of each other, their product is always −1.</p>

<p><strong>Worked Example:</strong> Find the slope of a line parallel to y = 4x − 3.</p>

<p>Parallel lines share the same slope: 4.</p>

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<p class="prompt">1. A line is parallel to y = 2x + 7. What is its slope?</p>

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<p><span class="label">Explanation:</span> Parallel lines have the same slope: 2.</p>

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<p class="prompt">2. A line is perpendicular to y = 2x + 7. What is its slope?</p>

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<p><span class="label">Explanation:</span> The perpendicular slope is the opposite reciprocal of 2, which is −<span class="frac"><span class="num">1</span><span class="den">2</span></span>.</p>

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<p class="prompt">3. Write the equation of a line parallel to y = 3x − 4 that passes through (2, 5).</p>

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<p><span class="label">Explanation:</span> Parallel means slope = 3. y − 5 = 3(x − 2), so y = 3x − 6 + 5 = 3x − 1.</p>

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<p class="prompt">4. Write the equation of a line perpendicular to y = −<span class="frac"><span class="num">1</span><span class="den">2</span></span>x + 6 that passes through (4, 3).</p>

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<p><span class="label">Explanation:</span> The perpendicular slope of −<span class="frac"><span class="num">1</span><span class="den">2</span></span> is 2. y − 3 = 2(x − 4), so y = 2x − 8 + 3 = 2x − 5.</p>

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<p class="prompt">5. Two lines have slopes <span class="frac"><span class="num">3</span><span class="den">4</span></span> and −<span class="frac"><span class="num">4</span><span class="den">3</span></span>. Are these lines parallel, perpendicular, or neither?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pc-5',true)">A) Perpendicular</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-5',false)">B) Parallel</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-5',false)">C) Neither</button>

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<p><span class="label">Explanation:</span> Their product is <span class="frac"><span class="num">3</span><span class="den">4</span></span> × (−<span class="frac"><span class="num">4</span><span class="den">3</span></span>) = −1, confirming the lines are perpendicular.</p>

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<h2 id="intercepts">4. Intercepts and Writing Equations from Points</h2>

<p>The <strong>x-intercept</strong> is where a line crosses the x-axis (set y = 0 and solve). The <strong>y-intercept</strong> 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.</p>

<p><strong>Worked Example:</strong> Find the x- and y-intercepts of 2x + 3y = 12.</p>

<p>x-intercept (set y = 0): 2x = 12, so x = 6, giving (6, 0).</p>

<p>y-intercept (set x = 0): 3y = 12, so y = 4, giving (0, 4).</p>

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<p class="prompt">1. Find the x-intercept of the line 4x − 5y = 20.</p>

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<p><span class="label">Explanation:</span> Set y = 0: 4x = 20, so x = 5.</p>

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<p class="prompt">2. Find the y-intercept of the line 4x − 5y = 20.</p>

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<p><span class="label">Explanation:</span> Set x = 0: −5y = 20, so y = −4.</p>

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<p class="prompt">3. A line has x-intercept 6 and y-intercept −3. What is its slope?</p>

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<p><span class="label">Explanation:</span> The intercepts are the points (6, 0) and (0, −3). Slope: <span class="frac"><span class="num">0 − (−3)</span><span class="den">6 − 0</span></span> = <span class="frac"><span class="num">3</span><span class="den">6</span></span> = <span class="frac"><span class="num">1</span><span class="den">2</span></span>.</p>

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<p class="prompt">4. A line passes through the points (0, 5) and (10, 0). What is its equation in slope-intercept form?</p>

</div>

<p><span class="label">Explanation:</span> Slope: <span class="frac"><span class="num">0 − 5</span><span class="den">10 − 0</span></span> = −<span class="frac"><span class="num">1</span><span class="den">2</span></span>. Since (0, 5) is already the y-intercept: y = −<span class="frac"><span class="num">1</span><span class="den">2</span></span>x + 5.</p>

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<h2 id="mistakes">Common Mistakes to Avoid</h2>

<li><strong>Subtracting the coordinates in the wrong order.</strong> 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.</li>

<li><strong>Mixing up which line type has slope 0 versus an undefined slope.</strong> A horizontal line has slope 0 (flat, no rise). A vertical line has undefined slope (infinite steepness, division by zero).</li>

<li><strong>Using the same slope for a perpendicular line.</strong> A perpendicular line needs the opposite reciprocal of the original slope, not the same slope (that would make it parallel instead).</li>

<li><strong>Forgetting to flip the sign when finding a reciprocal.</strong> The opposite reciprocal of <span class="frac"><span class="num">2</span><span class="den">3</span></span> is −<span class="frac"><span class="num">3</span><span class="den">2</span></span>, not <span class="frac"><span class="num">3</span><span class="den">2</span></span>, both the flip and the sign change matter.</li>

<li><strong>Forgetting to set the other variable to 0 when finding an intercept.</strong> 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.</li>

</ul>

<h2 id="faq">Frequently Asked Questions</h2>

<h3>Which form of a line's equation should I use?</h3>

<p>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.</p>

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<h3>How can I quickly tell if two lines are parallel or perpendicular from their equations?</h3>

<p>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.</p>

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<h3>Why does dividing by zero make a vertical line's slope undefined, instead of just very large?</h3>

<p>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.</p>

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<h3>Where can I practice more problems like these?</h3>

<p>The <a href="https://theschoolofmathematics.com/quiz/act-lines-quiz-1">Lines quizzes</a> 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.</p>

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/act-lines-quiz-1">Practice Lines Free</a>

<a class="cta-btn cta-secondary" href="https://theschoolofmathematics.com/quiz/course/ACT-Math-Qbank">Explore the Full ACT Math Qbank</a>

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