Question

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

X y

3 -5

4 -10

5 6

-15 -20

Answer 1

To find the linear equation that fits the given points, we can start by determining the slope (\(m\)) and y-intercept (\(b\)) of the line in the slope-intercept form, which is \(y = mx + b\).

The points given are:
1. (3, -5)
2. (4, -10)
3. (5, 6)
4. (-15, -20)

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

We can use two points to calculate the slope. Let's use the points (3, -5) and (5, 6):

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - (-5)}{5 - 3} = \frac{6 + 5}{5 - 3} = \frac{11}{2}
\]

Thus, the slope \(m\) is \(\frac{11}{2}\).

### Step 2: Use the slope and a point to find the y-intercept \(b\)

We can use the point (3, -5) to find \(b\):
\[
y = mx + b
\]
Substituting \(x = 3\) and \(y = -5\):
\[
-5 = \frac{11}{2}(3) + b
\]
Calculating:
\[
-5 = \frac{33}{2} + b
\]
Subtract \(\frac{33}{2}\) from both sides:
\[
b = -5 - \frac{33}{2} = -\frac{10}{2} - \frac{33}{2} = -\frac{43}{2}
\]

### Step 3: Write the final linear equation

Now we have the slope \(m = \frac{11}{2}\) and the y-intercept \(b = -\frac{43}{2}\). Hence, the linear equation is:

\[
y = \frac{11}{2}x - \frac{43}{2}
\]

So the linear equation that gives the rule for the table is:

\[
y = \frac{11}{2}x - \frac{43}{2}
\]