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) \( m \) of a linear function between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Here, rate of change is \( +5 \), and we can use points \((3,13)\) and \((4,a)\), or \((4,a)\) and \((5,23)\). Let's use \((3,13)\) and \((4,a)\): \( 5=\frac{a - 13}{4 - 3} \).

Step2: Solve for \( a \)

Simplify the denominator: \( 4 - 3 = 1 \). So the equation becomes \( 5=\frac{a - 13}{1} \), which simplifies to \( a - 13 = 5 \). Add 13 to both sides: \( a = 5 + 13 = 18 \). We can check with the other pair \((4,a)\) and \((5,23)\): \( 5=\frac{23 - a}{5 - 4}\Rightarrow 5=\frac{23 - a}{1}\Rightarrow 23 - a = 5\Rightarrow a = 23 - 5 = 18 \), which matches.

Answer:

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