A tailor charges a fixed fee for cutting a pattern plus an amount for every metre of fabric bought. The total cost, \(y\) pounds, for \(x\) metres of fabric...
Question 1 Report
A tailor charges a fixed fee for cutting a pattern plus an amount for every metre of fabric bought. The total cost, \(y\) pounds, for \(x\) metres of fabric follows a straight-line rule that passes through the points \((3, 26)\) and \((7, 42)\).
Find the equation of this line in the form \(y = mx + c\). (3)
Answer Details
The tailor's total cost is a straight-line rule, so knowing two points on that line is enough to find both the cost per metre (the gradient) and the fixed cutting fee (the intercept).
Using the two given points, \((3, 26)\) and \((7, 42)\), the gradient is:
\[\frac{42 - 26}{7 - 3} = \frac{16}{4} = 4 \quad \textbf{[1 mark]}\]
Substituting the point \((3, 26)\) and gradient 4 into \(y = mx + c\):
\[26 = 4(3) + c \implies 26 = 12 + c \implies c = 14 \quad \textbf{[1 mark]}\]
So the equation of the line is:
\[y = 4x + 14 \quad \textbf{[1 mark]}\]
Here the gradient, 4, is the cost per metre of fabric, and the intercept, 14, is the fixed fee for cutting the pattern.
Exam tip: when a straight-line rule is described only through two points, always find the gradient first using both points, then substitute one point back in to find the intercept.
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