Interpreting the Point of Intersection
Make sense of the point of intersection where two straight-line graphs meet for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-01). When two lines model two options, such as two phone plans or two gyms, the crossing point is where both options give the same value.
You will find the crossing point on a graph, explain what its \(x\)-value and \(y\)-value mean in words with units, and decide which option is cheaper on each side of it — in real-life settings such as taxi fares, tradespeople, savings and filling or draining tanks.
Theory
When two straight lines model two options, their point of intersection is where the options are equal. This Year 12 Standard 1 (NSW) guide shows how to read the crossing point, explain it in words, and decide which option is cheaper on each side — for savings, tradespeople, gyms and tanks.
In a practical problem each straight line models one option, for example \(\text{cost}=\text{fixed fee}+\text{rate}\times\text{amount}\). The point of intersection \((x,y)\) is where the two options give the same value.
The \(x\)-value (across) tells you when or how many; the \(y\)-value (up) is the amount — dollars, litres and so on. Always explain the point in words with units: “at 4 hours both shops charge \(\$24\)”.
On one side of the crossing one option is lower (cheaper); on the other side the options swap. Without a graph, find the crossing by making the two rules equal.
Each option is usually a straight-line rule:
At the point of intersection the two options are equal:
How to interpret a point of intersection
- Read the crossing point \((x,y)\) from the graph, or make the two rules equal and solve.
- Explain it in words with units: the \(x\)-value is when or how many, the \(y\)-value is the amount.
- Compare each side of the crossing: the lower line is the cheaper option, and it swaps at the crossing.
- Check by substituting the crossing value into both rules.
The lines cross at \((10,50)\). Check with both rules.
| \(\text{Lena: } 10+4(10)\) | \(=\) | \(\$50\) |
| \(\text{Tom: } 30+2(10)\) | \(=\) | \(\$50\) |
\((10,50)\): after 10 weeks they both have \(\$50\).
Let \(h\) = hours. The rules are \(C=60+40h\) (A) and \(C=80h\) (B). Substitute \(h=3\) into both.
| \(\text{A: } C\) | \(=\) | \(60+40(3)=\$180\) |
| \(\text{B: } C\) | \(=\) | \(80(3)=\$240\) |
\(\$180<\$240\), and 3 hours is past the crossing, so Electrician A is cheaper (by \(\$60\)).
Read the crossing: \((5,50)\). Check with the rules \(V=100-10t\) and \(V=20+6t\).
| \(\text{P: } 100-10(5)\) | \(=\) | \(50\) |
| \(\text{Q: } 20+6(5)\) | \(=\) | \(50\) |
After 5 minutes both tanks hold 50 L.
(a) Make the two costs equal. (b) Substitute \(v=14\) into both.
| \(50+10v\) | \(=\) | \(15v\) |
| \(50\) | \(=\) | \(5v\) |
| \(v\) | \(=\) | \(10\) |
| \(\text{X: } 50+10(14)\) | \(=\) | \(\$190\) |
| \(\text{Y: } 15(14)\) | \(=\) | \(\$210\) |
Both cost \(50+10(10)=15(10)=\$150\). (a) 10 visits (\(\$150\)); (b) Gym X is cheaper.
Common pitfalls
Frequently asked questions
What does the point of intersection mean in a word problem?
It is where the two options give the same value. The x-value tells you when or how many, and the y-value tells you the amount, such as the cost in dollars. For example, at 4 hours both shops charge 24 dollars.
How do you tell which option is cheaper from a graph?
The cheaper option is the lower line. On one side of the crossing one line is lower, and on the other side the lines swap, so check which side your value is on.
How do you find the point of intersection without a graph?
Make the two rules equal and solve. For example, 16 plus 2n equals 6n gives n equals 4, and substituting back gives a cost of 24 dollars.
Which coordinate of the intersection answers the question?
It depends on the question. 'How many hours?' or 'when?' is the x-value on the horizontal axis. 'What cost?' or 'how much?' is the y-value on the vertical axis.
How do you compare two plans at a given amount?
Substitute the amount into both rules, or read both lines at that value, and compare the answers. The smaller cost is the better plan for that amount.