A reliable method
Name the unknowns and write one equation for each independent condition. Use substitution when a variable is already isolated; otherwise eliminate a variable by adding or subtracting equations.
The common trap
A money total and an item count require consistent units. Do not add dollars to tickets or treat a ten-cent coin as ten dollars.
Check and transfer
After solving, reconstruct both totals. Negative counts or a non-integer number of indivisible objects signal that either the model or the interpretation needs attention.
Worked comparison: why the alternative fails
Ten tickets earn $62 at $8 per adult and $5 per child. Ten child tickets would earn $50. The $12 extra represents four $3 upgrades, so there are 4 adults and 6 children. Check 4×8+6×5=62 as well as the count.
Transfer and explain
For two numbers totaling 18 with a difference of 6, the smaller is (18−6)/2 = 6. The larger is 12; dividing 18 equally would ignore the second condition.
A show sells 12 tickets: adult tickets at $8 and child tickets at $5. Revenue is $75. How many adult tickets were sold?
- If every ticket cost $5, calculate the baseline revenue.
- Each adult ticket adds $3 above that baseline; divide the extra revenue by 3.
- The requested value is 5. Check its units and meaning against the original question.
Compare the other answer choices
Report child tickets instead of adult tickets.
Divide total revenue by adult price as though all tickets cost the same, taking whole tickets.
Assume every ticket is an adult ticket.
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.
GED Mathematical Reasoning overviewSiyavula Grade 10 mathematics foundation reference