Question

How to find the slope of a line given an equation

Answer 1

To find the slope of a line given its equation, first, you need to identify the form of the equation. The most common forms are the slope-intercept form and the standard form. Here’s how to find the slope for each case:

### 1. Slope-Intercept Form

If the equation is in the form:
\[ y = mx + b \]
where:
- \( m \) is the slope,
- \( b \) is the y-intercept,

the slope is simply the coefficient of \( x \).

**Example:**
For the equation \( y = 3x + 2 \), the slope \( m \) is 3.

### 2. Standard Form

If the equation is in the standard form:
\[ Ax + By = C \]
where \( A \), \( B \), and \( C \) are constants, you can rearrange it to find the slope:

1. Solve the equation for \( y \):
\[ By = -Ax + C \]
\[ y = -\frac{A}{B}x + \frac{C}{B} \]

2. In this form, the slope \( m \) is \( -\frac{A}{B} \).

**Example:**
For the equation \( 2x + 3y = 6 \):
1. Rearranging gives:
\[ 3y = -2x + 6 \]
\[ y = -\frac{2}{3}x + 2 \]
2. Thus, the slope \( m \) is \( -\frac{2}{3} \).

### 3. Point-Slope Form

If you have a point and the slope, it might be in point-slope form:
\[ y - y_1 = m(x - x_1) \]
Where:
- \( m \) is the slope,
- \( (x_1, y_1) \) is a point on the line.

In point-slope form, the slope is explicitly stated as \( m \).

### Summary
- For \( y = mx + b \), the slope is \( m \).
- For \( Ax + By = C \), rearrange to find slope as \( -\frac{A}{B} \).
- For \( y - y_1 = m(x - x_1) \), the slope is \( m \).

By recognizing the form of the equation you have, you can easily extract the slope of the line.