Question

(1, 3)
(0, -2)

Use the graph to derive the equation of the line in slope-intercept form.

___

Answer 1

To derive the equation of the line in slope-intercept form (y = mx + b) from the two given points \((1, 3)\) and \((0, -2)\), we need to follow these steps:

1. **Identify the points**:
- Point 1: \((x_1, y_1) = (1, 3)\)
- Point 2: \((x_2, y_2) = (0, -2)\)

2. **Calculate the slope (m)**:
The slope \(m\) is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Plugging in our points:
\[
m = \frac{-2 - 3}{0 - 1} = \frac{-5}{-1} = 5
\]

3. **Use the slope-intercept form**:
We now have \(m = 5\), so the equation starts as:
\[
y = 5x + b
\]

4. **Find the y-intercept (b)**:
The y-intercept is the value of \(y\) when \(x = 0\). From the second point \((0, -2)\), we see that when \(x = 0\), \(y = -2\). Therefore:
\[
b = -2
\]

5. **Write the final equation**:
Substitute \(b\) into the equation:
\[
y = 5x - 2
\]

Thus, the equation of the line in slope-intercept form is:
\[
\boxed{y = 5x - 2}
\]

Answer 2

(3, 9)
(2, 6)
(1, 3)
(0, 0)

Use similar triangles to determine the slope of the line.

The slope is ___

Answer 3

To determine the slope of the line using the given points \((3, 9)\), \((2, 6)\), \((1, 3)\), and \((0, 0)\), we can observe that these points are likely to form similar triangles with a common slope.

1. **Identify two points**:
Let’s take the points \((1, 3)\) and \((0, 0)\) to find the slope.

2. **Calculate the change in \(y\) (rise)**:
\[
\text{Rise} = y_2 - y_1 = 3 - 0 = 3
\]

3. **Calculate the change in \(x\) (run)**:
\[
\text{Run} = x_2 - x_1 = 1 - 0 = 1
\]

4. **Calculate the slope (m)**:
The slope \(m\) is given by:
\[
m = \frac{\text{Rise}}{\text{Run}} = \frac{3}{1} = 3
\]

Now we can assign this slope to all of the pairs formed with the other points.

For example, let’s check the slope between points \((3, 9)\) and \((2, 6)\):
- **Change in \(y\)**:
\[
6 - 9 = -3
\]
- **Change in \(x\)**:
\[
2 - 3 = -1
\]
- **Slope**:
\[
m = \frac{-3}{-1} = 3
\]

Also, checking points \((2, 6)\) and \((1, 3)\) gives:
- **Change in \(y\)**:
\[
3 - 6 = -3
\]
- **Change in \(x\)**:
\[
1 - 2 = -1
\]
- **Slope**:
\[
m = \frac{-3}{-1} = 3
\]

Lastly, checking points \((1, 3)\) and \((0, 0)\):
- **Change in \(y\)**:
\[
3 - 0 = 3
\]
- **Change in \(x\)**:
\[
1 - 0 = 1
\]
- **Slope**:
\[
m = \frac{3}{1} = 3
\]

Since every set of points gives us the same slope, **the slope of the line is**:
\[
\boxed{3}
\]