To find the linear equation that represents the relationship between \(x\) and \(y\) in the given table, we can use the format of a linear equation:
\[ y = mx + b \]
where \(m\) is the slope and \(b\) is the y-intercept.
First, we will calculate the slope \(m\) using two points from the table:
- \( (5, 44) \)
- \( (10, 59) \)
The formula for the slope is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using our points:
\[ m = \frac{59 - 44}{10 - 5} = \frac{15}{5} = 3 \]
Now, we have the slope \(m = 3\).
Next, we'll use one of the points to solve for \(b\). Let's use the point \( (5, 44) \):
\[ 44 = 3(5) + b \] \[ 44 = 15 + b \] \[ b = 44 - 15 = 29 \]
Now, we can write the equation:
\[ y = 3x + 29 \]
So, the final linear equation is:
\[ y = 3x + 29 \]