Write the linear equation that gives the rule for this table.

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
36 & 8 \\
\hline
41 & 18 \\
\hline
46 & 28 \\
\hline
51 & 38 \\
\hline
\end{tabular}

Write your answer as an equation with [tex]\(y\)[/tex] first, followed by an equals sign.

[tex]\[\boxed{\text{y = }}\][/tex]



Answer :

To find the linear equation that defines the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] given the table, we need to determine the slope and the y-intercept.

1. Determine the Slope (m):
The slope [tex]\( m \)[/tex] is calculated using two points from the table. Let's use the points (36, 8) and (41, 18):
[tex]\[ m = \frac{y_1 - y_0}{x_1 - x_0} = \frac{18 - 8}{41 - 36} = \frac{10}{5} = 2.0 \][/tex]

2. Determine the y-intercept (b):
The y-intercept [tex]\( b \)[/tex] can be found using the formula [tex]\( y = mx + b \)[/tex]. Using the point (36, 8) and the slope [tex]\( m = 2.0 \)[/tex]:
[tex]\[ 8 = 2.0 \cdot 36 + b \implies 8 = 72 + b \implies b = 8 - 72 = -64.0 \][/tex]

3. Form the Equation:
With the slope [tex]\( m = 2.0 \)[/tex] and y-intercept [tex]\( b = -64.0 \)[/tex], the linear equation is:
[tex]\[ y = 2.0x - 64.0 \][/tex]

Therefore, the linear equation that defines the rule for the given table is:
[tex]\[ y = 2.0x - 64.0 \][/tex]