Which equation shows a line passing through the points $(4,28)$ and $(9,63)$?
To identify the line passing through the points \((4, 28)\) and \((9, 63)\), we first determine the slope of the line. The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \[m = \frac{y_2 - y_1}{x_2 - x_1}.\] Substituting the given points: \[m = \frac{63 - 28}{9 - 4} = \frac{35}{5} = 7.\] Now that we know the slope is 7, we test the given options to see which one is consistent with these points.
We test the options given:
Testing point \((4, 28)\) in the options:
Since only option (a) satisfies the first point, we verify it with the second point \((9, 63)\):
Thus, the equation of the line is \(y = 7x\).
In economics, linear relationships are often used to represent simple cost functions. If you know that producing 4 units costs 28 dollars and producing 9 units costs 63 dollars, assuming a constant rate, you can model the cost function as \(y = 7x\), where \(y\) is the total cost and \(x\) is the number of units. This allows you to predict the cost for any number of units, such as 10 units costing \(\$70\).