To determine the value of [tex]\(a\)[/tex] so that the data in the table represent a linear function with a rate of change (or slope) of +5, follow these steps:
1. Identify the given points:
- Point 1: [tex]\((3, 13)\)[/tex]
- Point 2: [tex]\((5, 23)\)[/tex]
2. Understand the linear function formula:
For a linear function, the rate of change (slope) between any two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
We are given that the rate of change is +5.
3. Verify the slope between the given points:
Use the points [tex]\((3, 13)\)[/tex] and [tex]\((5, 23)\)[/tex] to check the slope:
[tex]\[
\text{slope} = \frac{23 - 13}{5 - 3} = \frac{10}{2} = 5
\][/tex]
The given points confirm the slope of +5.
4. Find [tex]\((4, a)\)[/tex] using the slope:
Since the slope is constant (+5), we can use the point [tex]\((3, 13)\)[/tex] to find [tex]\(a\)[/tex] when [tex]\(x = 4\)[/tex].
5. Calculate [tex]\(a\)[/tex]:
We know the slope formula:
[tex]\[
\text{slope} = \frac{a - y_1}{4 - x_1}
\][/tex]
Plug in the known values [tex]\(y_1 = 13\)[/tex] and [tex]\(x_1 = 3\)[/tex], and the slope of 5:
[tex]\[
5 = \frac{a - 13}{4 - 3}
\][/tex]
Simplify the equation:
[tex]\[
5 = a - 13
\][/tex]
Solve for [tex]\(a\)[/tex]:
[tex]\[
a = 5 + 13 = 18
\][/tex]
Therefore, the value of [tex]\(a\)[/tex] that makes the data represent a linear function with a rate of change of +5 is [tex]\(\boxed{18}\)[/tex].
