Choose a method
Name the unknowns, build one equation per condition, and use substitution or elimination. Check your result in both original conditions.
Avoid this trap
Finding one variable does not finish a question that asks for the other variable or a combination. Read the requested expression before doing extra algebra.
Check and transfer
When the variable coefficients are proportional, compare the constants to distinguish infinitely many solutions from no solution. An entire equation that is a scaled copy describes the same line.
Worked example with a numerical check
A cafeteria sold 12 lunches, each either soup or pasta. Pasta lunches numbered twice the soup lunches. How many pasta lunches were sold? Represent soup by s and pasta by 2s. Their total is 3s, so find one part and multiply by two. The requested value is 8.
Contrast and diagnose
The equations 2u + v = 14 and u + v = 10 hold. What is u + 2v? Subtract the second equation from the first to find u. Find v from the simpler equation, then compute the requested combination. The requested value is 16. An incorrect approach: Stop at u instead of computing u+2v.
Transfer practice
In a system, y = x + 4 and y = 3x − 4. What is the common y-value? Try first. Check: 8. Set the two expressions for y equal and solve for x. Use either equation to recover y.
A cafeteria sold 12 lunches, each either soup or pasta. Pasta lunches numbered twice the soup lunches. How many pasta lunches were sold?
- Represent soup by s and pasta by 2s.
- Their total is 3s, so find one part and multiply by two.
- The requested value is 8.
Compare the other answer choices
Split into equal groups instead of a 1:2 ratio.
Report all lunches.
Double the entire total instead of the soup part.
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.
Math specifications