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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 [tex]\(a\)[/tex]?

[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
3 & 13 \\
\hline
4 & a \\
\hline
5 & 23 \\
\hline
\end{array}
\][/tex]

A. [tex]\(a=3\)[/tex]
B. [tex]\(a=8\)[/tex]
C. [tex]\(a=18\)[/tex]
D. [tex]\(a=33\)[/tex]


Sagot :

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].
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