ACT Math: System of Equations
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.
1. Solving Systems by Substitution
Substitution 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.
Worked Example: Solve the system y = 6x − 5 and 4x + 2y = 22.
4x + 2(6x − 5) = 22
4x + 12x − 10 = 22
16x = 32, so x = 2
Substitute back: y = 6(2) − 5 = 7. Solution: (2, 7).
1. Solve using substitution: y = 2x + 3 and 3x + y = 18
Explanation: 3x + (2x + 3) = 18, so 5x = 15, x = 3. y = 2(3) + 3 = 9.
2. Solve using substitution: x = y + 4 and 2x + 3y = 23
Explanation: 2(y + 4) + 3y = 23, so 2y + 8 + 3y = 23, 5y = 15, y = 3. x = 3 + 4 = 7.
3. Solve using substitution: y = 4x and x + y = 25
Explanation: x + 4x = 25, so 5x = 25, x = 5. y = 4(5) = 20.
4. Solve using substitution: 2x + y = 11 and y = x − 1
Explanation: 2x + (x − 1) = 11, so 3x = 12, x = 4. y = 4 − 1 = 3.
5. Solve using substitution: x = 3y − 2 and 2x + y = 17
Explanation: 2(3y − 2) + y = 17, so 6y − 4 + y = 17, 7y = 21, y = 3. x = 3(3) − 2 = 7.
2. Solving Systems by Elimination
Elimination works by adding or subtracting the two equations (after multiplying one or both by a constant, if needed) so that one variable cancels out.
Worked Example: Solve the system 3x + 2y = 16 and 3x − y = 1.
Subtract the second equation from the first, since both have 3x:
(3x + 2y) − (3x − y) = 16 − 1
3y = 15, so y = 5
Substitute back: 3x − 5 = 1, so 3x = 6 and x = 2. Solution: (2, 5).
1. Solve using elimination: x + y = 10 and x − y = 2
Explanation: Add the equations: 2x = 12, so x = 6. y = 10 − 6 = 4.
2. Solve using elimination: 2x + 3y = 13 and 2x − y = 1
Explanation: Subtract: (2x + 3y) − (2x − y) = 13 − 1, so 4y = 12, y = 3. Then 2x − 3 = 1, so x = 2.
3. Solve using elimination: 4x + y = 17 and x + y = 8
Explanation: Subtract: (4x + y) − (x + y) = 17 − 8, so 3x = 9, x = 3. y = 8 − 3 = 5.
4. Solve using elimination: 3x + 2y = 12 and x + y = 5
Explanation: 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.
5. Solve using elimination: 5x − 2y = 16 and 3x + 2y = 16
Explanation: Add the equations: 8x = 32, so x = 4. Then 3(4) + 2y = 16, so 2y = 4 and y = 2.
3. Graphing and Classifying Solutions
Graphically, a system's solution is where the two lines intersect. A linear system always has exactly one of three outcomes:
| Slopes and intercepts | Graph | Number of solutions |
|---|---|---|
| Different slopes | Lines cross once | Exactly 1 |
| Same slope, different y-intercepts | Parallel lines | 0 |
| Same slope, same y-intercept | The same line | Infinitely many |
1 solution
0 solutions (parallel)
Worked Example: How many solutions does the system y = 2x + 3 and y = 2x − 1 have?
Both lines have slope 2, but different y-intercepts (3 and −1), so they're parallel and never intersect.
Answer: 0 solutions.
1. How many solutions does the system y = 3x + 5 and y = 3x + 5 have?
Explanation: Both equations describe the exact same line, so every point on it is a solution: infinitely many.
2. How many solutions does the system y = 4x − 2 and y = −4x + 6 have?
Explanation: The slopes (4 and −4) are different, so the lines cross at exactly one point.
3. How many solutions does the system 2x + y = 7 and 4x + 2y = 10 have?
Explanation: 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.
4. A system of two linear equations has the same slope and the same y-intercept. How many solutions does it have?
Explanation: Matching slope and y-intercept means the two equations describe the same line, giving infinitely many shared points.
5. On a graph, two lines intersect at exactly one point, (3, −2). What is the solution to the system?
Explanation: The point where the two lines cross is, by definition, the solution to the system: (3, −2).
4. Word Problems
Systems of equations translate real situations with two unknowns and two relationships into a solvable pair of equations.
Worked Example: 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.)
Let c = chickens, w = cows.
c + w = 30 2c + 4w = 74
Divide the second equation by 2: c + 2w = 37. Subtract the first equation:
(c + 2w) − (c + w) = 37 − 30, so w = 7
Then c = 30 − 7 = 23 chickens.
1. The sum of two numbers is 45, and their difference is 11. What is the larger number?
Explanation: Let x and y be the numbers: x + y = 45 and x − y = 11. Adding: 2x = 56, so x = 28.
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?
Explanation: 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.
3. Two numbers have a sum of 20. Twice the first number minus the second number equals 13. What are the two numbers?
Explanation: Let x + y = 20 and 2x − y = 13. Adding: 3x = 33, so x = 11 and y = 9.
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.)
Explanation: 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.
Common Mistakes to Avoid
- Substituting into the same equation you solved. After isolating a variable in one equation, substitute the resulting expression into the other equation, not back into the same one, or you'll just get a true but useless statement.
- Adding when you should subtract, or vice versa, during elimination. Check whether the matching terms have the same sign (subtract to eliminate) or opposite signs (add to eliminate) before combining the equations.
- Forgetting to solve for the second variable. A system's solution is an ordered pair; after finding one variable, always substitute back to find the other.
- Assuming every system has exactly one solution. Parallel lines produce no solution, and identical lines produce infinitely many, always check the slopes and intercepts before assuming a unique answer exists.
- Setting up a word problem's two equations from the same piece of information. 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.
Frequently Asked Questions
How do I decide between substitution and elimination?
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.
What does it mean for a system to have "no solution"?
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.
How can I check my answer to a system of equations?
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.
Where can I practice more problems like these?
The System of Equations 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.