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

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
2 & 58 \\
\hline
3 & 65 \\
\hline
4 & 72 \\
\hline
5 & 79 \\
\hline
\end{tabular}

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

[tex]$\square$[/tex]



Answer :

To determine the linear equation that describes the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] based on the given table, we follow these steps:

1. Identify two points from the table:
Given the points are:
[tex]\[ (2, 58), (3, 65), (4, 72), (5, 79) \][/tex]

2. Calculate the slope (m):
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]
Using the points [tex]\((2, 58)\)[/tex] and [tex]\((5, 79)\)[/tex]:
[tex]\[ m = \frac{79 - 58}{5 - 2} = \frac{21}{3} = 7.0 \][/tex]

3. Calculate the y-intercept (b):
The formula for the y-intercept [tex]\( b \)[/tex] when the slope [tex]\( m \)[/tex] and a point [tex]\((x_1, y_1)\)[/tex] are known is:
[tex]\[ b = y_1 - m \cdot x_1 \][/tex]
Using the point [tex]\((2, 58)\)[/tex] and the slope [tex]\( m = 7.0 \)[/tex]:
[tex]\[ b = 58 - (7.0 \cdot 2) = 58 - 14 = 44.0 \][/tex]

4. Form the linear equation:
Using the slope [tex]\( m = 7.0 \)[/tex] and the y-intercept [tex]\( b = 44.0 \)[/tex], the linear equation in the form [tex]\( y = mx + b \)[/tex] is:
[tex]\[ y = 7.0x + 44.0 \][/tex]

So, the linear equation that describes the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the given table is:
[tex]\[ y = 7.0x + 44.0 \][/tex]