In week 2 a family spent £180 on groceries, and in week 5 they spent £240. Grocery spending increases at a constant rate with the week number. The family ke...
Question 1 Report
In week 2 a family spent £180 on groceries, and in week 5 they spent £240. Grocery spending increases at a constant rate with the week number. The family keeps careful records to help plan future weekly shopping within a tight budget.
- Find the gradient of the line joining these two points. (1)
- Find the equation of the line in the form \(y = mx + c\). (2)
Answer Details
Two points on a straight line are enough to find both its gradient and its full equation. The gradient is the change in \(y\) divided by the change in \(x\) between the points; the equation then follows by substituting one point into \(y - y_1 = m(x - x_1)\).
- Using \((2, 180)\) and \((5, 240)\): \[m = \dfrac{240 - 180}{5 - 2} = \dfrac{60}{3} = 20\] The gradient is \(20\) pounds per week. [1]
- Substituting \((2, 180)\) and \(m = 20\) into \(y - y_1 = m(x - x_1)\): \[y - 180 = 20(x - 2)\] \[y = 20x - 40 + 180 = 20x + 140\] [2] (correct method, \(y = 20x + 140\))
Check an equation like this by substituting the other known point back in: at \(x=5\), \(y = 20(5) + 140 = 240\), which matches the given data and confirms the line is correct.
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