To determine the missing value in the table for the linear function, we need to find the linear equation that describes the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. The linear equation of a function is generally written in the form:
[tex]\[ y = mx + b \][/tex]
where:
- [tex]\( m \)[/tex] is the slope of the line.
- [tex]\( b \)[/tex] is the y-intercept.
To find the slope [tex]\( m \)[/tex], we will use two known points on the line. Let's use the points [tex]\((4, 10)\)[/tex] and [tex]\((12, 16)\)[/tex].
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the given points [tex]\((4, 10)\)[/tex] and [tex]\((12, 16)\)[/tex]:
[tex]\[ m = \frac{16 - 10}{12 - 4} = \frac{6}{8} = 0.75 \][/tex]
Next, we need to find the y-intercept [tex]\( b \)[/tex]. We can use the equation of the line and one of the known points. Let's use the point [tex]\((4, 10)\)[/tex]:
[tex]\[ y = mx + b \][/tex]
[tex]\[ 10 = 0.75 \cdot 4 + b \][/tex]
[tex]\[ 10 = 3 + b \][/tex]
[tex]\[ b = 10 - 3 \][/tex]
[tex]\[ b = 7 \][/tex]
Now we have the equation of the line:
[tex]\[ y = 0.75x + 7 \][/tex]
To find the missing value at [tex]\( x = 9 \)[/tex], we substitute [tex]\( x = 9 \)[/tex] into the equation:
[tex]\[ y = 0.75 \cdot 9 + 7 \][/tex]
[tex]\[ y = 6.75 + 7 \][/tex]
[tex]\[ y = 13.75 \][/tex]
Therefore, the missing value [tex]\( y \)[/tex] when [tex]\( x = 9 \)[/tex] is:
[tex]\[ \boxed{13.75} \][/tex]
