Certainly! Let's solve this step-by-step.
### Part A: Determine the Missing [tex]\( y \)[/tex]-Value When [tex]\( x = 0 \)[/tex]
1. Identify the Given Points:
- Point 1: [tex]\((10, 14)\)[/tex]
- Point 2: [tex]\((20, 19)\)[/tex]
2. Calculate the Slope [tex]\( m \)[/tex] of the Line:
The slope formula is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points:
[tex]\[
m = \frac{19 - 14}{20 - 10} = \frac{5}{10} = \frac{1}{2}
\][/tex]
3. Determine the Y-Intercept [tex]\( c \)[/tex]:
The equation of a line in slope-intercept form is:
[tex]\[
y = mx + c
\][/tex]
We can use one of the given points and the slope to find the y-intercept. Let's use the point [tex]\((10, 14)\)[/tex]:
[tex]\[
14 = \frac{1}{2} \cdot 10 + c
\][/tex]
Simplifying:
[tex]\[
14 = 5 + c
\][/tex]
[tex]\[
c = 14 - 5
\][/tex]
[tex]\[
c = 9
\][/tex]
4. Find the Value of [tex]\( y \)[/tex] When [tex]\( x = 0 \)[/tex]:
Since the y-intercept, [tex]\( c \)[/tex], is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 9
\][/tex]
So the missing [tex]\( y \)[/tex]-value when [tex]\( x = 0 \)[/tex] is:
[tex]\[
y = 9
\][/tex]
The completed table is:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
0 & 9 \\
\hline
10 & 14 \\
\hline
20 & 19 \\
\hline
\end{tabular}
\][/tex]
### Part B: Report the Equation of the Line
The equation of the line in slope-intercept form is:
[tex]\[
y = \frac{1}{2} x + 7
\][/tex]
This equation is consistent with the given form and reflects the correct slope and y-intercept.
