To find the equation of the linear function in the form \( y = mx + b \), we need to determine the slope \( m \) and the y-intercept \( b \) using the points provided in the table.
The two points given are:
- \( (6, 16) \)
- \( (15, 43) \)
Step 1: Calculate the slope \( m \)
The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the values from our points:
\[ m = \frac{43 - 16}{15 - 6} = \frac{27}{9} = 3 \]
Step 2: Use the slope-point form to find the equation
We now have the slope \( m = 3 \). We can use one of the points to find \( b \). Let's use the point \( (6, 16) \).
Substituting the values into the equation \( y = mx + b \):
\[ 16 = 3(6) + b \]
\[ 16 = 18 + b \]
Now, solve for \( b \):
\[ b = 16 - 18 = -2 \]
Step 3: Write the equation in slope-intercept form
Now we have both \( m \) and \( b \):
\[ y = 3x - 2 \]
So, the equation of the linear function that models the relationship shown in the table is:
\[ y = 3x - 2 \]
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