Mastering Line Equations: Understanding Slope-Intercept Form

How do you find the slope-intercept form?

To find the slope-intercept form of a linear equation, you need two key pieces of information: the slope ($m$) and the y-intercept ($b$). The slope-intercept form of a linear equation is given by:

$y = mx + b$

Here's how to find the slope-intercept form:

Step 1: Find the Slope ($m$)

The slope ($m$) represents the rate of change of the line and can be calculated using the following formula:

$m = \frac{{y_2 - y_1}}{{x_2 - x_1}}$

Where $(x_1, y_1)$ and $(x_2, y_2)$ are any two points on the line. You can choose any two points as long as they are distinct (i.e., not the same point).

Step 2: Find the Y-Intercept ($b$)

The y-intercept ($b$) is the point where the line intersects the y-axis. To find $b$, you can use one of the following methods:

• Method 1: Use a Known Point: If you have a point $(x_1, y_1)$ on the line (which you used to calculate the slope), you can rearrange the equation as follows to solve for $b$:

  $b = y_1 - mx_1$

• Method 2: Use the Y-Intercept: If you know that the line passes through the point $(0, b)$, then $b$ is simply the y-coordinate of the y-intercept.

Step 3: Write the Equation

Once you have found the slope ($m$) and the y-intercept ($b$), you can plug these values into the slope-intercept form equation:

$y = mx + b$

This equation now represents the linear relationship between $x$ and $y$ in slope-intercept form.

Let's go through an example:

Example: Find the Slope-Intercept Form

Suppose you have two points on a line: $(2, 4)$ and $(5, 7)$. You want to find the slope-intercept form of the line.

Step 1: Find the Slope ($m$)$m = \frac{{7 - 4}}{{5 - 2}} = \frac{3}{3} = 1$

Step 2: Find the Y-Intercept ($b$)You can use either of the methods mentioned earlier. Let's use Method 1:

$b = 4 - 1(2) = 4 - 2 = 2$

Step 3: Write the EquationNow, you have the slope ($m$) and the y-intercept ($b$). Plug them into the slope-intercept form equation:

$y = 1x + 2$

So, the slope-intercept form of the line passing through the points $(2, 4)$ and $(5, 7)$ is $y = x + 2$.

This equation represents the linear relationship between $x$ and $y$ for that line.