Question

Find the equation of the linear function represented by the table below in slope-intercept form.
x y
11 minus, 7−7
22 minus, 12−12
33 minus, 17−17
44 minus, 22−22

Answer 1

To find the equation of a linear function in slope-intercept form \( y = mx + b \), we first need to determine the slope \( m \) from the values given in the table:

| x | y |
|----|----|
| 11 | -7 |
| 22 | -12|
| 33 | -17|
| 44 | -22|

**Step 1: Calculate the slope \( m \)**

The slope \( m \) is determined by the formula:

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

We can choose any two points. Let's choose the first two points: \( (11, -7) \) and \( (22, -12) \).

So, using these points:

- \( (x_1, y_1) = (11, -7) \)
- \( (x_2, y_2) = (22, -12) \)

Calculating the slope:

\[
m = \frac{-12 - (-7)}{22 - 11} = \frac{-12 + 7}{11} = \frac{-5}{11}
\]

**Step 2: Use the slope to find the equation**

Now that we know \( m = -\frac{5}{11} \), we can use one of the points to find \( b \). Let’s use the point \( (11, -7) \).

The general form is:

\[
y = mx + b
\]

Substituting \( y = -7 \), \( m = -\frac{5}{11} \), and \( x = 11 \):

\[
-7 = -\frac{5}{11}(11) + b
\]

Calculating \( -\frac{5}{11} \times 11 = -5 \):

\[
-7 = -5 + b
\]

Now, solve for \( b \):

\[
b = -7 + 5 = -2
\]

**Step 3: Write the final equation**

Now we can write the equation in slope-intercept form:

\[
y = -\frac{5}{11}x - 2
\]

This is the equation that represents the linear function based on the data provided in the table.