Question

Write the equation of the line represented in the table. You must show ALL work to earn full credit.

x= 3,6,9,12,15
y=5,3,1,-1,-3

Answer 1

don't explain it just show me the steps

Answer 2

To find the equation of the line represented by the given values of \(x\) and \(y\), we need to determine the slope and the y-intercept.

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

The slope \(m\) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
We can pick any two points from the table. For our calculations, let's use the first two points: \((3, 5)\) and \((6, 3)\).

Using the coordinates:
- \(x_1 = 3\), \(y_1 = 5\)
- \(x_2 = 6\), \(y_2 = 3\)

Now plug these values into the slope formula:
\[
m = \frac{3 - 5}{6 - 3} = \frac{-2}{3}
\]

### Step 2: Use the slope-intercept form of the equation

The slope-intercept form of a line is given by the equation:
\[
y = mx + b
\]
where \(m\) is the slope and \(b\) is the y-intercept.

We now know that \(m = -\frac{2}{3}\). We will use one of the points to find \(b\). Let's use the point \((3, 5)\).

Substituting \(m\) and the coordinates into the equation:
\[
5 = -\frac{2}{3}(3) + b
\]

Now calculate \(-\frac{2}{3}(3)\):
\[
-\frac{2}{3}(3) = -2
\]

So we rewrite the equation:
\[
5 = -2 + b
\]

To find \(b\), add 2 to both sides:
\[
5 + 2 = b
\]
\[
b = 7
\]

### Step 3: Write the final equation

Now we have both \(m\) and \(b\):
- Slope \(m = -\frac{2}{3}\)
- Y-intercept \(b = 7\)

The equation of the line is:
\[
y = -\frac{2}{3}x + 7
\]

### Final Answer

Thus, the equation of the line represented by the given data is:
\[
\boxed{y = -\frac{2}{3}x + 7}
\]

Answer 3

Sure! Here are the steps without explanations:

### Step 1: Calculate the slope (m)
- Using points \((3, 5)\) and \((6, 3)\):
\[
m = \frac{3 - 5}{6 - 3} = \frac{-2}{3}
\]

### Step 2: Use the slope-intercept form
- Start with the equation:
\[
y = mx + b
\]
- Using point \((3, 5)\):
\[
5 = -\frac{2}{3}(3) + b
\]
\[
5 = -2 + b
\]
\[
b = 7
\]

### Step 3: Write the final equation
- The equation of the line:
\[
y = -\frac{2}{3}x + 7
\]

### Final Answer
\[
\boxed{y = -\frac{2}{3}x + 7}
\]