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Assume the table represents a linear function.

a. Complete the box in the first row.
\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
0 & [tex]$\square$[/tex] \\
\hline
10 & 14 \\
\hline
20 & 19 \\
\hline
\end{tabular}

b. Report the equation of the line in slope-intercept form: [tex]$y = \frac{1}{2} x + 7$[/tex]

(Simplify your answer. Type your answer in slope-intercept form. Use integers, fractions, or decimals.)


Sagot :

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.
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