A reliable method
Use substitution when one variable is already isolated. Use elimination when adding or subtracting equations removes a variable cleanly.
The common trap
After solving for one variable, return to an original equation to find the other. A ticket count, distance or time may need an additional nonnegative restriction.
Make the connection
Check both equations. In a cost comparison, a break-even time solves equality; whether one plan is cheaper before or after it depends on the rates.
Worked example: choose the requested quantity
The equations x + y = 10 and x − y = 4 hold. What is y? Subtract the second equation from the first to eliminate x. The resulting equation has 2y on the left; divide by 2. The requested value is 3.
Contrast and diagnose an error
A venue sells 7 tickets: standard tickets cost $12 and student tickets cost $7. Revenue is $64. How many standard tickets were sold? If all tickets were student tickets, compute the baseline revenue. Each standard ticket adds $5 above that baseline; divide the excess revenue by $5. The requested value is 3. A tempting answer is 15: Report extra dollars rather than ticket count.
Transfer practice and check
Plan A charges $4 plus $3 per hour. Plan B charges $10 plus $1 per hour. At how many hours do the charges match? Try it before reading on. Check: 3. Set the two total-cost expressions equal. The initial difference is recovered at a rate of $2 per hour.
The equations x + y = 10 and x − y = 4 hold. What is y?
- Subtract the second equation from the first to eliminate x.
- The resulting equation has 2y on the left; divide by 2.
- The requested value is 3.
Compare the other answer choices
Report x−y instead of y.
Add the equations and report x.
Subtract the equations in reverse order without changing the sign.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
ACT Mathematics content