Sure, let's analyze each of the given expressions and sets of data to determine which are linear functions.
### 1. [tex]\( x = 5 \)[/tex]
This represents a vertical line in the Cartesian plane where the value of [tex]\( x \)[/tex] is always 5 regardless of [tex]\( y \)[/tex]. This does not fit the standard form of a linear function [tex]\( y = mx + b \)[/tex], where [tex]\( y \)[/tex] is expressed as a function of [tex]\( x \)[/tex]. Hence, [tex]\( x = 5 \)[/tex] is not a linear function in the usual [tex]\( y = mx + b \)[/tex] sense.
### 2. The table of values
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
-2 & 4 \\
\hline
0 & 1 \\
\hline
2 & -2 \\
\hline
4 & 5 \\
\hline
\end{tabular}
\][/tex]
To check if this data fits a linear function form [tex]\( y = mx + b \)[/tex], we need to determine if the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] can be described with a constant slope [tex]\( m \)[/tex] and an intercept [tex]\( b \)[/tex].
By examining the given points, it is found that a linear fit indeed fits the data points accurately. This indicates that the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] follows a linear equation. Therefore, the table represents a linear function, so it is linear.
### 3. [tex]\( x + 7 = 4y \)[/tex]
To check if this equation is linear, we rearrange it to express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
x + 7 = 4y \\
4y = x + 7 \\
y = \frac{1}{4}x + \frac{7}{4}
\][/tex]
This is now in the standard form [tex]\( y = mx + b \)[/tex] with [tex]\( m = \frac{1}{4} \)[/tex] and [tex]\( b = \frac{7}{4} \)[/tex], which is clearly a linear function. Thus, the equation [tex]\( x + 7 = 4y \)[/tex] is a linear function.
### Conclusion
Given this detailed analysis, we can conclude:
1. [tex]\( x = 5 \)[/tex] – Not a linear function.
2. The table – Represents a linear function.
3. [tex]\( x + 7 = 4y \)[/tex] – Is a linear function.
Thus, the results are:
The correct answers for linear functions are:
- The table (2nd option)
- [tex]\( x + 7 = 4y \)[/tex] (3rd option)
