Question

This table shows the relationship between x and y values that will contribute to linear functions.
x
y

6 16
15 43
Write the equation of the linear function that models the relationship shown in the table. Enter your answer in y=mx+b
form.

Answer 1

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 given points \((6, 16)\) and \((15, 43)\).

### 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 given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

In our case, \((x_1, y_1) = (6, 16)\) and \((x_2, y_2) = (15, 43)\).

Plugging in the values:

\[
m = \frac{43 - 16}{15 - 6} = \frac{27}{9} = 3
\]

### Step 2: Use the slope-point form to find \(b\)

Now, we can use the slope and one of the points to solve for \(b\). We'll use the point \((6, 16)\).

The equation in slope-intercept form is:

\[
y = mx + b
\]

Substituting \(m = 3\), \(x = 6\), and \(y = 16\):

\[
16 = 3(6) + b
\]
\[
16 = 18 + b
\]

Now, solve for \(b\):

\[
b = 16 - 18 = -2
\]

### Step 3: Write the final equation

Now we have both \(m\) and \(b\):

\[
y = 3x - 2
\]

Thus, the equation of the linear function that models the relationship shown in the table is:

\[
\boxed{y = 3x - 2}
\]