System of Equations | Free Act Math Course

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<h1>ACT Math: System of Equations</h1>

A system of equations asks for the values that satisfy two (or more) equations simultaneously. This free, complete lesson covers solving systems by substitution, solving by elimination, graphing a system and classifying its solution as one, none, or infinitely many, and real-world word problems that translate into a system of two equations. 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-system-of-equations-quiz-1">Practice System of Equations 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="#substitution">Solving Systems by Substitution</a></li>

<li><a href="#elimination">Solving Systems by Elimination</a></li>

<li><a href="#graphing">Graphing and Classifying Solutions</a></li>

<li><a href="#word-problems">Word Problems</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="substitution">1. Solving Systems by Substitution</h2>

<p><strong>Substitution</strong> works best when one equation is already solved for a variable, or can be solved for one easily. Substitute that expression into the other equation, solve for the remaining variable, then substitute back to find the first.</p>

<p><strong>Worked Example:</strong> Solve the system y = 6x − 5 and 4x + 2y = 22.</p>

<p>Substitute back: y = 6(2) − 5 = 7. Solution: (2, 7).</p>

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<p class="prompt">1. Solve using substitution: y = 2x + 3 and 3x + y = 18</p>

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<p><span class="label">Explanation:</span> 3x + (2x + 3) = 18, so 5x = 15, x = 3. y = 2(3) + 3 = 9.</p>

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<p class="prompt">2. Solve using substitution: x = y + 4 and 2x + 3y = 23</p>

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<p><span class="label">Explanation:</span> 2(y + 4) + 3y = 23, so 2y + 8 + 3y = 23, 5y = 15, y = 3. x = 3 + 4 = 7.</p>

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<p class="prompt">3. Solve using substitution: y = 4x and x + y = 25</p>

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<p><span class="label">Explanation:</span> x + 4x = 25, so 5x = 25, x = 5. y = 4(5) = 20.</p>

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<p class="prompt">4. Solve using substitution: 2x + y = 11 and y = x − 1</p>

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<p><span class="label">Explanation:</span> 2x + (x − 1) = 11, so 3x = 12, x = 4. y = 4 − 1 = 3.</p>

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<p class="prompt">5. Solve using substitution: x = 3y − 2 and 2x + y = 17</p>

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<p><span class="label">Explanation:</span> 2(3y − 2) + y = 17, so 6y − 4 + y = 17, 7y = 21, y = 3. x = 3(3) − 2 = 7.</p>

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<h2 id="elimination">2. Solving Systems by Elimination</h2>

<p><strong>Elimination</strong> works by adding or subtracting the two equations (after multiplying one or both by a constant, if needed) so that one variable cancels out.</p>

<p><strong>Worked Example:</strong> Solve the system 3x + 2y = 16 and 3x − y = 1.</p>

<p>Subtract the second equation from the first, since both have 3x:</p>

<p>Substitute back: 3x − 5 = 1, so 3x = 6 and x = 2. Solution: (2, 5).</p>

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<p class="prompt">1. Solve using elimination: x + y = 10 and x − y = 2</p>

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<p><span class="label">Explanation:</span> Add the equations: 2x = 12, so x = 6. y = 10 − 6 = 4.</p>

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<p class="prompt">2. Solve using elimination: 2x + 3y = 13 and 2x − y = 1</p>

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<p><span class="label">Explanation:</span> Subtract: (2x + 3y) − (2x − y) = 13 − 1, so 4y = 12, y = 3. Then 2x − 3 = 1, so x = 2.</p>

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<p class="prompt">3. Solve using elimination: 4x + y = 17 and x + y = 8</p>

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<p><span class="label">Explanation:</span> Subtract: (4x + y) − (x + y) = 17 − 8, so 3x = 9, x = 3. y = 8 − 3 = 5.</p>

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<p class="prompt">4. Solve using elimination: 3x + 2y = 12 and x + y = 5</p>

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<p><span class="label">Explanation:</span> Multiply the second equation by 2: 2x + 2y = 10. Subtract from the first: (3x + 2y) − (2x + 2y) = 12 − 10, so x = 2. Then 2 + y = 5, so y = 3.</p>

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<p class="prompt">5. Solve using elimination: 5x − 2y = 16 and 3x + 2y = 16</p>

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<p><span class="label">Explanation:</span> Add the equations: 8x = 32, so x = 4. Then 3(4) + 2y = 16, so 2y = 4 and y = 2.</p>

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<h2 id="graphing">3. Graphing and Classifying Solutions</h2>

<p>Graphically, a system's solution is where the two lines intersect. A linear system always has exactly one of three outcomes:</p>

<tr><th>Slopes and intercepts</th><th>Graph</th><th>Number of solutions</th></tr>

<tr><td>Different slopes</td><td>Lines cross once</td><td>Exactly 1</td></tr>

<tr><td>Same slope, different y-intercepts</td><td>Parallel lines</td><td>0</td></tr>

<tr><td>Same slope, same y-intercept</td><td>The same line</td><td>Infinitely many</td></tr>

</table>

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<p class="figure-caption">1 solution</p>

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<p class="figure-caption">0 solutions (parallel)</p>

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<p><strong>Worked Example:</strong> How many solutions does the system y = 2x + 3 and y = 2x − 1 have?</p>

<p>Both lines have slope 2, but different y-intercepts (3 and −1), so they're parallel and never intersect.</p>

<p>Answer: 0 solutions.</p>

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<p class="prompt">1. How many solutions does the system y = 3x + 5 and y = 3x + 5 have?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pc-1',true)">A) Infinitely many</button>

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<p><span class="label">Explanation:</span> Both equations describe the exact same line, so every point on it is a solution: infinitely many.</p>

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<p class="prompt">2. How many solutions does the system y = 4x − 2 and y = −4x + 6 have?</p>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-2',false)">C) Infinitely many</button>

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<p><span class="label">Explanation:</span> The slopes (4 and −4) are different, so the lines cross at exactly one point.</p>

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<p class="prompt">3. How many solutions does the system 2x + y = 7 and 4x + 2y = 10 have?</p>

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

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<p><span class="label">Explanation:</span> Rewrite both in slope-intercept form: y = −2x + 7 and y = −2x + 5. Same slope, different y-intercepts, so the lines are parallel: 0 solutions.</p>

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<p class="prompt">4. A system of two linear equations has the same slope and the same y-intercept. How many solutions does it have?</p>

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

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-4',false)">D) Cannot be determined</button>

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<p><span class="label">Explanation:</span> Matching slope and y-intercept means the two equations describe the same line, giving infinitely many shared points.</p>

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<p class="prompt">5. On a graph, two lines intersect at exactly one point, (3, −2). What is the solution to the system?</p>

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<p><span class="label">Explanation:</span> The point where the two lines cross is, by definition, the solution to the system: (3, −2).</p>

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<h2 id="word-problems">4. Word Problems</h2>

<p>Systems of equations translate real situations with two unknowns and two relationships into a solvable pair of equations.</p>

<p><strong>Worked Example:</strong> A farm has chickens and cows totaling 30 animals with 74 legs total. How many chickens are there? (Chickens have 2 legs, cows have 4.)</p>

<p>Let c = chickens, w = cows.</p>

<p>Divide the second equation by 2: c + 2w = 37. Subtract the first equation:</p>

<p>Then c = 30 − 7 = 23 chickens.</p>

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<p class="prompt">1. The sum of two numbers is 45, and their difference is 11. What is the larger number?</p>

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<p><span class="label">Explanation:</span> Let x and y be the numbers: x + y = 45 and x − y = 11. Adding: 2x = 56, so x = 28.</p>

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<p class="prompt">2. A movie theater sells adult tickets for $12 and child tickets for $8. If 150 tickets were sold for a total of $1,560, how many adult tickets were sold?</p>

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<p><span class="label">Explanation:</span> Let a + c = 150 and 12a + 8c = 1,560. Substitute c = 150 − a: 12a + 8(150 − a) = 1,560, giving 12a + 1,200 − 8a = 1,560, so 4a = 360 and a = 90.</p>

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<p class="prompt">3. Two numbers have a sum of 20. Twice the first number minus the second number equals 13. What are the two numbers?</p>

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<p><span class="label">Explanation:</span> Let x + y = 20 and 2x − y = 13. Adding: 3x = 33, so x = 11 and y = 9.</p>

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<p class="prompt">4. A pet store has parrots and rabbits totaling 24 animals with 76 legs total. How many rabbits are there? (Parrots have 2 legs, rabbits have 4.)</p>

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<p><span class="label">Explanation:</span> Let p + r = 24 and 2p + 4r = 76. Divide the second by 2: p + 2r = 38. Subtract the first: (p + 2r) − (p + r) = 38 − 24, so r = 14.</p>

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

<li><strong>Substituting into the same equation you solved.</strong> After isolating a variable in one equation, substitute the resulting expression into the <em>other</em> equation, not back into the same one, or you'll just get a true but useless statement.</li>

<li><strong>Adding when you should subtract, or vice versa, during elimination.</strong> Check whether the matching terms have the same sign (subtract to eliminate) or opposite signs (add to eliminate) before combining the equations.</li>

<li><strong>Forgetting to solve for the second variable.</strong> A system's solution is an ordered pair; after finding one variable, always substitute back to find the other.</li>

<li><strong>Assuming every system has exactly one solution.</strong> Parallel lines produce no solution, and identical lines produce infinitely many, always check the slopes and intercepts before assuming a unique answer exists.</li>

<li><strong>Setting up a word problem's two equations from the same piece of information.</strong> Each equation in the system must capture a genuinely different relationship between the variables, reusing the same fact twice won't produce a solvable system.</li>

</ul>

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

<h3>How do I decide between substitution and elimination?</h3>

<p>Use substitution when one equation is already solved for a variable, or a variable has a coefficient of 1, making it easy to isolate. Use elimination when both equations are in standard form (Ax + By = C) and the coefficients of one variable are the same or easy to match by multiplying.</p>

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<h3>What does it mean for a system to have "no solution"?</h3>

<p>It means there is no pair of values that satisfies both equations at once, graphically, the two lines never intersect because they're parallel. Algebraically, trying to solve such a system leads to a false statement, like 5 = 8.</p>

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<h3>How can I check my answer to a system of equations?</h3>

<p>Substitute both values back into both original equations. If they make both equations true, the solution is correct, this quick check catches most arithmetic slips.</p>

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

<p>The <a href="https://theschoolofmathematics.com/quiz/act-system-of-equations-quiz-1">System of Equations 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-system-of-equations-quiz-1">Practice System of Equations 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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