Linear Relationships

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<h1>PSAT/NMSQT Math: Linear Relationships</h1>

Linear relationships questions ask you to read, build, and interpret lines, whether they're given as equations, tables, graphs, or word descriptions. This free, complete lesson covers slope and rate of change, slope-intercept and point-slope form, parallel and perpendicular lines, and interpreting graphs, intercepts, and intersections in real-world context. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quizzes below.

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/psat-linear-relationships-quiz-1">Practice Linear Relationships Free</a>

<a class="cta-btn cta-secondary" href="https://theschoolofmathematics.com/quiz/psat-linear-relationships-quiz-2">Try Linear Relationships Quiz 2</a>

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

<ol>

<li><a href="#slope">Slope and Rate of Change</a></li>

<li><a href="#forms">Slope-Intercept and Point-Slope Form</a></li>

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

<li><a href="#context">Interpreting Graphs, Intercepts, and Intersections</a></li>

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

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

</ol>

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<h2 id="slope">1. Slope and Rate of Change</h2>

<p>The <strong>slope</strong> of a line measures how steeply it rises or falls, defined as the change in y divided by the change in x between any two points on the line: m = (y₂ − y₁)/(x₂ − x₁). In real-world contexts, slope represents a constant <strong>rate of change</strong>, such as dollars per hour or miles per gallon.</p>

<p><strong>Worked Example:</strong> Find the slope of the line through (2, 3) and (6, 11).</p>

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

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

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<p class="prompt">2. A taxi charges a $4 base fee plus $2.50 per mile. What does the rate of change represent in this context?</p>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p1-2',false)">C) The total number of miles</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p1-2',false)">D) The total cost of the ride</button>

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<p><span class="label">Explanation:</span> Rate of change is the slope, the amount the cost changes per mile, which is $2.50; the base fee is the y-intercept, a fixed starting value.</p>

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<p class="prompt">3. A line passes through (0, 5) and has a slope of 0. Which statement is true?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'p1-3',true)">A) The line is horizontal, and y = 5 for every x</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p1-3',false)">B) The line is vertical</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p1-3',false)">C) The line passes through the origin</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p1-3',false)">D) x = 5 for every y</button>

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<p><span class="label">Explanation:</span> A slope of 0 means y never changes as x changes, producing a horizontal line at y = 5.</p>

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<p class="prompt">4. A gym's membership data shows that from 100 members it grew by 15 members every month, at a constant rate. If m represents the number of months since the start, which expression gives the number of members?</p>

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<p><span class="label">Explanation:</span> The starting value 100 is the constant term, and the constant rate of change 15 per month is the slope multiplying m.</p>

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<h2 id="forms">2. Slope-Intercept and Point-Slope Form</h2>

<p><strong>Slope-intercept form</strong>, y = mx + b, directly shows the slope m and the y-intercept b. <strong>Point-slope form</strong>, y − y₁ = m(x − x₁), is useful when you know a point on the line and its slope but not yet the y-intercept, and can always be rearranged into slope-intercept form.</p>

<p><strong>Worked Example:</strong> Write the equation, in slope-intercept form, of the line through (4, 1) with slope 3.</p>

<p>Point-slope: y − 1 = 3(x − 4)</p>

</div>

<p class="prompt">1. Write the equation of the line with slope 2 that passes through (0, −5) in slope-intercept form.</p>

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<p><span class="label">Explanation:</span> Since (0, −5) is the y-intercept, b = −5, so y = 2x − 5.</p>

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<p class="prompt">2. A line passes through (2, 7) with slope −1. Write its equation in point-slope form.</p>

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<p><span class="label">Explanation:</span> Substituting (x₁, y₁) = (2, 7) and m = −1 into y − y₁ = m(x − x₁) gives y − 7 = −1(x − 2).</p>

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<p class="prompt">3. Convert y − 3 = 4(x − 2) into slope-intercept form.</p>

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<p><span class="label">Explanation:</span> y − 3 = 4x − 8 → y = 4x − 5.</p>

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<p class="prompt">4. Find the slope and y-intercept of the line 3x + 2y = 12.</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'p2-4',true)">A) slope = −3/2, y-intercept = 6</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p2-4',false)">B) slope = 3/2, y-intercept = 6</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p2-4',false)">C) slope = −3/2, y-intercept = 12</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p2-4',false)">D) slope = 3, y-intercept = 2</button>

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<p><span class="label">Explanation:</span> Solving for y: 2y = −3x + 12 → y = −<span class="frac"><span class="num">3</span><span class="den">2</span></span>x + 6, so the slope is −3/2 and the y-intercept is 6.</p>

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

<p><strong>Parallel lines</strong> have the exact same slope and never intersect. <strong>Perpendicular lines</strong> intersect at a right angle, and their slopes are <strong>negative reciprocals</strong> of each other, meaning they multiply to −1.</p>

<p><strong>Worked Example:</strong> Find the slope of a line perpendicular to y = −<span class="frac"><span class="num">2</span><span class="den">3</span></span>x + 4.</p>

<p>The given slope is −2/3. The negative reciprocal flips and negates it:</p>

<p class="step-math">perpendicular slope = 3/2</p>

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<p class="prompt">1. A line is parallel to y = 5x − 2 and passes through (0, 3). What is its equation?</p>

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<p><span class="label">Explanation:</span> Parallel lines share the same slope, 5, and the new y-intercept is 3, giving y = 5x + 3.</p>

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<p class="prompt">2. What is the slope of any line perpendicular to y = 4x + 1?</p>

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<p><span class="label">Explanation:</span> The negative reciprocal of 4 is −1/4.</p>

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<p class="prompt">3. Line A has slope 2/3 and line B has slope −3/2. What is the relationship between the lines?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'p3-3',true)">A) Perpendicular</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p3-3',false)">B) Parallel</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p3-3',false)">D) Cannot be determined</button>

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<p><span class="label">Explanation:</span> (2/3)(−3/2) = −1, confirming the slopes are negative reciprocals, so the lines are perpendicular.</p>

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<p class="prompt">4. A line perpendicular to y = −<span class="frac"><span class="num">1</span><span class="den">2</span></span>x + 7 passes through (0, −3). What is its equation?</p>

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<p><span class="label">Explanation:</span> The negative reciprocal of −1/2 is 2, and the y-intercept is −3, giving y = 2x − 3.</p>

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<h2 id="context">4. Interpreting Graphs, Intercepts, and Intersections</h2>

<p>On the graph of a line, the <strong>y-intercept</strong> is where the line crosses the y-axis (where x = 0), and the <strong>x-intercept</strong> is where it crosses the x-axis (where y = 0). Where two lines cross is the single point that satisfies both equations at once, and this point can be read directly off a graph or found algebraically.</p>

<p><strong>Worked Example:</strong> Find the x-intercept of y = 2x − 8.</p>

<p class="step-math">0 = 2x − 8 → x = 4, so the x-intercept is (4, 0)</p>

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<p class="prompt">1. Find the x-intercept of the line 3x + 4y = 24.</p>

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<p><span class="label">Explanation:</span> Set y = 0: 3x = 24 → x = 8, so the x-intercept is (8, 0).</p>

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<p class="prompt">2. A graph shows two lines that appear to cross at the point (3, 5). What does this point represent?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'p4-2',true)">A) The one (x, y) pair that satisfies both equations at the same time</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p4-2',false)">B) The y-intercept of both lines</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p4-2',false)">C) The slope of both lines</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p4-2',false)">D) A point that satisfies neither equation</button>

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<p><span class="label">Explanation:</span> Wherever two graphed lines intersect, that single (x, y) point is the solution that makes both equations true simultaneously.</p>

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<p class="prompt">3. A company's profit is modeled by P = 8u − 240, where u is the number of units sold. What does the y-intercept, −240, represent?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'p4-3',true)">A) A fixed cost of $240 incurred even if no units are sold</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p4-3',false)">B) The profit earned per unit sold</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p4-3',false)">C) The number of units that must be sold to break even</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p4-3',false)">D) The maximum possible profit</button>

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<p><span class="label">Explanation:</span> When u = 0, P = −240, meaning the company starts at a $240 loss before selling anything, a fixed cost.</p>

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<p class="prompt">4. Using P = 8u − 240 from the previous problem, how many units must be sold to break even (P = 0)?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'p4-4',true)">A) 30 units</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p4-4',false)">B) 8 units</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p4-4',false)">C) 240 units</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p4-4',false)">D) 32 units</button>

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<p><span class="label">Explanation:</span> 0 = 8u − 240 → 8u = 240 → u = 30 units, the x-intercept of the graph.</p>

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<p class="prompt">5. A graph shows a line passing through the points (0, 4) and (2, 4). What can you conclude?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'p4-5',true)">A) The line is horizontal with slope 0 and equation y = 4</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p4-5',false)">B) The line is vertical with equation x = 4</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p4-5',false)">C) The line has an undefined slope</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'p4-5',false)">D) The line passes through the origin</button>

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<p><span class="label">Explanation:</span> Since y stays at 4 for two different x-values, the slope is 0 and the line is the horizontal line y = 4.</p>

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

<li><strong>Swapping the numerator and denominator in the slope formula.</strong> Slope is change in y over change in x, (y₂ − y₁)/(x₂ − x₁), never the reverse.</li>

<li><strong>Confusing parallel and perpendicular slope rules.</strong> Parallel lines share the exact same slope; perpendicular lines have slopes that are negative reciprocals (they multiply to −1).</li>

<li><strong>Mixing up which value is the slope and which is the y-intercept</strong> when an equation isn't already in y = mx + b form. Always isolate y first.</li>

<li><strong>Setting the wrong variable to zero when finding an intercept.</strong> Set y = 0 for the x-intercept, and set x = 0 for the y-intercept.</li>

<li><strong>Misreading what a slope or intercept represents in a word problem.</strong> The slope is always a rate of change per unit, and the y-intercept is always the starting or fixed value when the input is zero.</li>

</ul>

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

<h3>How do I find the equation of a line from just two points?</h3>

<p>First calculate the slope using the slope formula, then substitute the slope and either point into point-slope form, y − y₁ = m(x − x₁), and simplify into slope-intercept form if needed.</p>

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<h3>Do parallel and perpendicular lines ever have the same y-intercept?</h3>

<p>Yes, that's possible for either relationship. What matters for parallel lines is that the slopes match, and for perpendicular lines, that the slopes are negative reciprocals; the y-intercepts can be anything, including equal or different.</p>

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<h3>What's the fastest way to find an x-intercept or y-intercept?</h3>

<p>Substitute 0 for the other variable and solve. Setting y = 0 gives the x-intercept, and setting x = 0 gives the y-intercept, no need to graph anything.</p>

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

<p>The <a href="https://theschoolofmathematics.com/quiz/psat-linear-relationships-quiz-1">Linear Relationships quizzes</a> in the PSAT/NMSQT Math Question Bank include additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT/NMSQT. You can also browse the <a href="https://theschoolofmathematics.com/quiz/course/PSAT-NMSQT-QBank/practice">full PSAT/NMSQT Math Question Bank</a>.</p>

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/psat-linear-relationships-quiz-1">Practice Linear Relationships Free</a>

<a class="cta-btn cta-secondary" href="https://theschoolofmathematics.com/quiz/psat-linear-relationships-quiz-2">Try Linear Relationships Quiz 2</a>

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