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To determine whether the given expressions and points represent linear functions, let's analyze each one step-by-step.
### 1. [tex]\(x = 5\)[/tex]
The equation [tex]\( x = 5 \)[/tex] is a vertical line in the [tex]\( (x, y) \)[/tex] coordinate plane, where the value of [tex]\( x \)[/tex] is always 5 irrespective of [tex]\( y \)[/tex]. For a function to be linear in the conventional sense [tex]\( y = mx + b \)[/tex], any such line needs to have a slope defined by [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] and [tex]\( b \)[/tex] are constants. The equation [tex]\( x = 5 \)[/tex] does not meet this requirement as the value of [tex]\( x \)[/tex] doesn't depend on [tex]\( y \)[/tex]. Therefore, [tex]\( x = 5 \)[/tex] is not considered a linear function in this context.
### 2. The Table of Points
The given data points are:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -2 & 4 \\ \hline 0 & 1 \\ \hline 2 & -2 \\ \hline 4 & 5 \\ \hline \end{array} \][/tex]
We need to check if these points lie on a line of the form [tex]\( y = mx + b \)[/tex]. To determine if the points form a linear function, we can calculate the slope [tex]\( m \)[/tex] between each pair of points and check if it remains constant.
Let's compute the slope [tex]\( m \)[/tex] between consecutive points:
- Slope between [tex]\( (-2, 4) \)[/tex] and [tex]\( (0, 1) \)[/tex]:
[tex]\[ m = \frac{1 - 4}{0 - (-2)} = \frac{-3}{2} = -1.5 \][/tex]
- Slope between [tex]\( (0, 1) \)[/tex] and [tex]\( (2, -2) \)[/tex]:
[tex]\[ m = \frac{-2 - 1}{2 - 0} = \frac{-3}{2} = -1.5 \][/tex]
- Slope between [tex]\( (2, -2) \)[/tex] and [tex]\( (4, 5) \)[/tex]:
[tex]\[ m = \frac{5 - (-2)}{4 - 2} = \frac{7}{2} = 3.5 \][/tex]
The slopes between consecutive points are [tex]\( -1.5 \)[/tex], [tex]\( -1.5 \)[/tex], and [tex]\( 3.5 \)[/tex]. Since the slopes are not consistent, these points do not form a linear function.
### 3. [tex]\( x + 7 = 4y \)[/tex]
We can rewrite [tex]\( x + 7 = 4y \)[/tex] to see if it fits the form [tex]\( y = mx + b \)[/tex]:
[tex]\[ x + 7 = 4y \implies 4y = x + 7 \implies y = \frac{1}{4}x + \frac{7}{4} \][/tex]
This equation is of the form [tex]\( y = mx + b \)[/tex], where the slope [tex]\( m = \frac{1}{4} \)[/tex] and the intercept [tex]\( b = \frac{7}{4} \)[/tex]. Hence, [tex]\( x + 7 = 4y \)[/tex] is indeed a linear function.
### Conclusion
Based on our analysis:
- [tex]\( x = 5 \)[/tex] is not a linear function.
- The given set of data points do not represent a linear function.
- [tex]\( x + 7 = 4y \)[/tex] is a linear function.
Thus, the only linear function from the given options is [tex]\( x + 7 = 4y \)[/tex].
### 1. [tex]\(x = 5\)[/tex]
The equation [tex]\( x = 5 \)[/tex] is a vertical line in the [tex]\( (x, y) \)[/tex] coordinate plane, where the value of [tex]\( x \)[/tex] is always 5 irrespective of [tex]\( y \)[/tex]. For a function to be linear in the conventional sense [tex]\( y = mx + b \)[/tex], any such line needs to have a slope defined by [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] and [tex]\( b \)[/tex] are constants. The equation [tex]\( x = 5 \)[/tex] does not meet this requirement as the value of [tex]\( x \)[/tex] doesn't depend on [tex]\( y \)[/tex]. Therefore, [tex]\( x = 5 \)[/tex] is not considered a linear function in this context.
### 2. The Table of Points
The given data points are:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -2 & 4 \\ \hline 0 & 1 \\ \hline 2 & -2 \\ \hline 4 & 5 \\ \hline \end{array} \][/tex]
We need to check if these points lie on a line of the form [tex]\( y = mx + b \)[/tex]. To determine if the points form a linear function, we can calculate the slope [tex]\( m \)[/tex] between each pair of points and check if it remains constant.
Let's compute the slope [tex]\( m \)[/tex] between consecutive points:
- Slope between [tex]\( (-2, 4) \)[/tex] and [tex]\( (0, 1) \)[/tex]:
[tex]\[ m = \frac{1 - 4}{0 - (-2)} = \frac{-3}{2} = -1.5 \][/tex]
- Slope between [tex]\( (0, 1) \)[/tex] and [tex]\( (2, -2) \)[/tex]:
[tex]\[ m = \frac{-2 - 1}{2 - 0} = \frac{-3}{2} = -1.5 \][/tex]
- Slope between [tex]\( (2, -2) \)[/tex] and [tex]\( (4, 5) \)[/tex]:
[tex]\[ m = \frac{5 - (-2)}{4 - 2} = \frac{7}{2} = 3.5 \][/tex]
The slopes between consecutive points are [tex]\( -1.5 \)[/tex], [tex]\( -1.5 \)[/tex], and [tex]\( 3.5 \)[/tex]. Since the slopes are not consistent, these points do not form a linear function.
### 3. [tex]\( x + 7 = 4y \)[/tex]
We can rewrite [tex]\( x + 7 = 4y \)[/tex] to see if it fits the form [tex]\( y = mx + b \)[/tex]:
[tex]\[ x + 7 = 4y \implies 4y = x + 7 \implies y = \frac{1}{4}x + \frac{7}{4} \][/tex]
This equation is of the form [tex]\( y = mx + b \)[/tex], where the slope [tex]\( m = \frac{1}{4} \)[/tex] and the intercept [tex]\( b = \frac{7}{4} \)[/tex]. Hence, [tex]\( x + 7 = 4y \)[/tex] is indeed a linear function.
### Conclusion
Based on our analysis:
- [tex]\( x = 5 \)[/tex] is not a linear function.
- The given set of data points do not represent a linear function.
- [tex]\( x + 7 = 4y \)[/tex] is a linear function.
Thus, the only linear function from the given options is [tex]\( x + 7 = 4y \)[/tex].
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