The intersection of two lines is the pair of values that makes both equations true at once. In a real problem, that pair usually marks a break-even point, a meeting point or a moment when two quantities are equal.
This lesson sits after the two solving methods in simultaneous relationships. You already know how to get x and y. Here you learn what to write afterwards.
How do you turn an equation into a sentence?
Every equation in context is built from a fixed part and a changing part. In C = 30 + 5n, the 30 is a fixed charge and the 5 is the cost for each extra visit.
When you solve a pair such as C = 30 + 5n and C = 10 + 9n, you are asking: “For which n do the two costs match?” The answer is that n, together with the shared cost.
The method, step by step
- Name the letters in words, with units (n = number of visits, C = cost in RM).
- Write one equation per situation.
- Set them equal if both are already “C = …”, or use elimination or substitution otherwise.
- Solve, then find the second value.
- Check in both equations.
- Write a sentence that uses the context, the numbers and their units.
- Compare beyond the crossing point if the question asks which option is better.
Worked example
A gym offers two plans. Plan A charges RM30 to join plus RM5 per visit, and Plan B charges RM10 to join plus RM9 per visit.
For how many visits do both plans cost the same, and which is cheaper after that?
Step 1, define: n = visits, C = total cost in RM.
Step 2, equations: A: C = 30 + 5n. B: C = 10 + 9n.
Step 3, set equal: 30 + 5n = 10 + 9n, so 20 = 4n and n = 5.
Step 4, cost: A gives 30 + 25 = 55. B gives 10 + 45 = 55. Both equal RM55.
Step 5, sentence: After 5 visits, both plans cost the same, RM55.
Step 6, beyond the crossing: B rises RM9 per visit and A rises RM5, so B grows faster. For more than 5 visits, Plan A is cheaper. A check at 10 visits confirms it: A costs RM80 and B costs RM100.
The mistake to watch for
The usual slip is stopping at the number.
Mistaken answer: “n = 5.”
This does not say what n is, does not give the shared cost, and does not answer “which is cheaper”.
A full-mark answer names the quantity and the units, and answers the question that was asked. A second slip is to claim that Plan B is cheaper because its joining fee is lower. That is only true before the crossing point, and here it stops being true after 5 visits.
Check yourself
Try these on paper, then open each answer.
1. Plan P costs C = 15 + 2t and Plan Q costs C = 5 + 3t, where t is the number of hours and C is in RM. When are they equal, and which is cheaper for 20 hours?
Show answer
Set equal: 15 + 2t = 5 + 3t, so t = 10. The cost is 15 + 20 = 35. Check Q: 5 + 30 = 35. At 20 hours, P costs 15 + 40 = 55 and Q costs 5 + 60 = 65, so P is cheaper.
Both cost RM35 at 10 hours; P is cheaper at 20 hours.
2. Tank A holds 40 litres and drains 4 litres per minute. Tank B holds 10 litres and fills 2 litres per minute. When do they hold equal amounts, and how much?
Show answer
V = 40 − 4t and V = 10 + 2t. Set equal: 40 − 4t = 10 + 2t, so 30 = 6t and t = 5. Then V = 10 + 10 = 20. Check A: 40 − 20 = 20.
After 5 minutes, both tanks hold 20 litres.
3. The lines y = 3x + 6 and y = x + 2 describe two savings plans, where x is the number of weeks from now. The solution is x = −2, y = 0. Is this a sensible answer in context?
Show answer
Setting equal: 3x + 6 = x + 2, so 2x = −4 and x = −2. Check: 3(−2) + 6 = 0 and −2 + 2 = 0. A negative x would mean two weeks in the past, so the lines cross before the plans start.
The algebra is right, but the crossing is outside the useful range, so the plans never have equal savings from now on.
Where this leads next
The next step is solving a linear and quadratic pair, where the crossing can happen at two points. You can also practise everything in the simultaneous relationships practice set. The non-calculator working trainer is useful for checking the arithmetic in cost comparisons.
If your algebra is right but your final sentence loses marks, a teacher in online one-to-one Mathematics tuition can mark your wording against a model sentence and show what is missing.