Question

x | y
3 | 13
4 | a
5 | 23
in order for the data in the table to represent a linear function with a rate of change of +5, what must be the value of a?
○ a = 3
○ a = 8
○ a = 18
○ a = 33

Explanation:

Step1: Recall rate of change formula

The rate of change (slope) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(\frac{y_2 - y_1}{x_2 - x_1}\). For a linear function, the rate of change is constant. Here, rate of change is \(+5\), and we can use the points \((3, 13)\) and \((4, a)\).

Step2: Apply rate of change formula

Using the formula \(\frac{a - 13}{4 - 3}=5\) (since \(x_2 - x_1 = 4 - 3 = 1\) and rate of change is 5). Simplify the denominator: \(\frac{a - 13}{1}=5\).

Step3: Solve for \(a\)

Since \(\frac{a - 13}{1}=a - 13\), we have \(a - 13 = 5\). Add 13 to both sides: \(a = 5 + 13 = 18\). We can also check with the next point \((4, a)\) and \((5, 23)\): \(\frac{23 - a}{5 - 4}=5\), so \(23 - a = 5\), which gives \(a = 23 - 5 = 18\), confirming the solution.

Answer:

\(a = 18\) (corresponding to the option: \(a = 18\))