Question

Slope from a table
Slope from a graph
Slope from an equation

Answer 1

Certainly! Let's go through how to determine the slope from a table, a graph, and an equation.

### 1. Slope from a Table
To find the slope using a table, you need to look for pairs of (x, y) values. The slope (m) can be calculated using the formula:

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

Here’s how to do it step-by-step:

- Choose two points from the table, say \((x_1, y_1)\) and \((x_2, y_2)\).
- Subtract the y-values and divide by the difference of the x-values.

**Example Table:**

| x | y |
|---|---|
| 1 | 2 |
| 3 | 6 |

**Calculation:**

Using the points (1, 2) and (3, 6),

\[
m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2
\]

### 2. Slope from a Graph
To find the slope from a graph, follow these steps:

1. Identify two points on the line. These points should have integer coordinates for simplicity.
2. Use the same formula as above:

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

3. Count the vertical change (rise) and horizontal change (run) between these two points.

**Example:**
- If you use points (2, 3) and (5, 7):
- Rise = 7 - 3 = 4
- Run = 5 - 2 = 3

**Calculation:**

\[
m = \frac{4}{3}
\]

### 3. Slope from an Equation
When given a linear equation in the slope-intercept form, \(y = mx + b\), the slope \(m\) is directly given by the coefficient of \(x\).

**Example:**
For the equation:

\[
y = 2x + 3
\]

The slope \(m\) is 2.

If the equation is in standard form \(Ax + By = C\), you can convert it to slope-intercept form. The slope can be determined as follows:

1. Solve for \(y\):
\[
By = -Ax + C \implies y = -\frac{A}{B}x + \frac{C}{B}
\]
2. Here, the slope \(m\) is \(-\frac{A}{B}\).

**Example Equation:**
For the equation:

\[
3x + 2y = 6
\]

Converting to slope-intercept form:

\[
2y = -3x + 6 \implies y = -\frac{3}{2}x + 3
\]

The slope \(m = -\frac{3}{2}\).

### Summary
- **From a Table**: Calculate using two points.
- **From a Graph**: Identify points, calculate rise and run.
- **From an Equation**: Extract directly from slope-intercept form or convert from standard form.

Feel free to ask if you need clarifications or more examples!