# Writing Linear Equations When Given 2 Points

Writing an equation in the form \( y = mx + b \) when given a pair of coordinates that the line passes through involves a few simple steps. Let's break it down:

1. **Understand the Form**: In the equation \( y = mx + b \):

- \( m \) represents the slope of the line.

- \( b \) represents the y-intercept, which is the point where the line crosses the y-axis.

2. **Identify the Given Coordinates**: You're given a pair of coordinates \((x_1, y_1)\) that the line passes through. These coordinates will help us find the slope (\( m \)).

3. **Calculate the Slope (\( m \))**: The slope \( m \) can be calculated using the formula:

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

You can choose any other point on the line as \((x_2, y_2)\), but the one given to you is typically denoted by \((x_1, y_1)\).

4. **Substitute the Slope (\( m \)) and the Given Coordinates into the Equation**: After finding the slope, plug it along with one of the given coordinates \((x_1, y_1)\) into the equation \( y = mx + b \).

5. **Solve for the y-intercept (\( b \))**: Once you've substituted the slope (\( m \)) and the coordinates \((x_1, y_1)\) into the equation, you can solve for \( b \).

6. **Write the Equation**: Finally, once you've found both the slope \( m \) and the y-intercept \( b \), you can write the equation in the form \( y = mx + b \).

Let's summarize these steps with an example:

**Example:**

Suppose you're given the point \((3, 5)\) that the line passes through. Follow these steps to find the equation:

1. **Identify the Given Coordinates**: \( (x_1, y_1) = (3, 5) \)

2. **Calculate the Slope (\( m \))**: Using the formula:

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

Let's say we choose another point on the line, for example, \((7, 9)\):

\[ m = \frac{{9 - 5}}{{7 - 3}} = \frac{4}{4} = 1 \]

So, the slope \( m = 1 \).

3. **Substitute into the Equation**: Using the slope \( m = 1 \) and the given point \((3, 5)\), we have:

\[ y = 1 \cdot x + b \]

4. **Solve for the y-intercept (\( b \))**: Substitute the point \( (3, 5) \) into the equation:

\[ 5 = 1 \cdot 3 + b \]

\[ 5 = 3 + b \]

\[ b = 5 - 3 \]

\[ b = 2 \]

5. **Write the Equation**: Now, plug the slope \( m = 1 \) and the y-intercept \( b = 2 \) back into the equation:

\[ y = 1x + 2 \]

or simply:

\[ y = x + 2 \]

So, the equation of the line passing through the point \( (3, 5) \) is \( y = x + 2 \).

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