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) of a linear function between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(\frac{y_2 - y_1}{x_2 - x_1}\). Here, the rate of change is \(+ 5\). Let's take the points \((3,13)\) and \((4,a)\). The rate of change between these two points should be \(5\). So, \(\frac{a - 13}{4 - 3}=5\).

Step2: Solve for \(a\)

Simplify the denominator: \(4 - 3 = 1\). So the equation becomes \(\frac{a - 13}{1}=5\), which simplifies to \(a-13 = 5\). Adding \(13\) to both sides, we get \(a=5 + 13=18\). We can also check with the next pair \((4,a)\) and \((5,23)\). The rate of change should be \(5\), so \(\frac{23 - a}{5 - 4}=5\), which is \(\frac{23 - a}{1}=5\), so \(23 - a=5\), and \(a = 23 - 5=18\).

Answer:

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