To determine if the equation [tex]\(x + 7 = 4y\)[/tex] represents a linear function, let's first rearrange it into the standard form of a linear equation, which is [tex]\(y = mx + b\)[/tex].
Given the equation:
[tex]\[ x + 7 = 4y \][/tex]
First, solve for [tex]\(y\)[/tex]:
[tex]\[ 4y = x + 7 \][/tex]
[tex]\[ y = \frac{1}{4}x + \frac{7}{4} \][/tex]
This is now in the form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept. Since it can be written in this form, the equation represents a linear function.
Now, let's analyze the table of values to see if they fit this linear function [tex]\(y = \frac{1}{4}x + \frac{7}{4}\)[/tex].
The given table is:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
2 & 4 \\
\hline
91 & 1 \\
\hline
2 & M \\
\hline
41 & \\
\hline
\end{array}
\][/tex]
1. Checking the first pair [tex]\((2, 4)\)[/tex]:
Substitute [tex]\(x = 2\)[/tex] into the equation [tex]\(y = \frac{1}{4}x + \frac{7}{4}\)[/tex]:
[tex]\[
y = \frac{1}{4}(2) + \frac{7}{4} = \frac{2}{4} + \frac{7}{4} = \frac{9}{4} = 2.25
\][/tex]
The [tex]\(y\)[/tex] value does not match with the given [tex]\(y = 4\)[/tex]. Hence, the point [tex]\((2, 4)\)[/tex] does not lie on the line defined by [tex]\(x + 7 = 4y\)[/tex].
2. Checking the second pair [tex]\((91, 1)\)[/tex]:
Substitute [tex]\(x = 91\)[/tex] into the equation [tex]\(y = \frac{1}{4}x + \frac{7}{4}\)[/tex]:
[tex]\[
y = \frac{1}{4}(91) + \frac{7}{4} = \frac{91}{4} + \frac{7}{4} = \frac{98}{4} = 24.5
\][/tex]
The [tex]\(y\)[/tex] value does not match with the given [tex]\(y = 1\)[/tex]. Hence, the point [tex]\((91, 1)\)[/tex] does not lie on the line defined by [tex]\(x + 7 = 4y\)[/tex].
3. Checking the pair [tex]\((41, )\)[/tex]:
Substitute [tex]\(x = 41\)[/tex] into the equation [tex]\(y = \frac{1}{4}x + \frac{7}{4}\)[/tex]:
[tex]\[
y = \frac{1}{4}(41) + \frac{7}{4} = \frac{41}{4} + \frac{7}{4} = \frac{48}{4} = 12
\][/tex]
The [tex]\(y\)[/tex] value should be [tex]\(12\)[/tex] for [tex]\(x = 41\)[/tex].
In summary, while we cannot verify point [tex]\((2, M)\)[/tex], the remaining points [tex]\((2, 4)\)[/tex] and [tex]\((91, 1)\)[/tex] do not fit the linear function equation derived from [tex]\(x + 7 = 4y\)[/tex]. Thus, based on the given data, none of the points presented in the table satisfy the equation [tex]\(x + 7 = 4y\)[/tex] when checked individually.
