The Slope Intercept and General Forms of a Line

# The Slope Intercept and General Forms of a Line

Before we look at the various forms a line can take, we will first familiarize ourselves with what the slope of a line is. Formally, the slope (commonly denoted with the letter $m$) is defined to be "rise over run", that is how many points up/down we must move over how many points left/right we must move in order to go from any point to another on the line. For example, the following graph has a slope where we go up $3$ and go to the right $2$, so we say the slope is $m = \frac{3}{2}$:

If the numerator is positive, we are rising up, while if the numerator is negative, we are rising down. Similarly, if the denominator is positive, we are running right, while if the denominator is negative, we are running left.

Another way to calculate slope is by taking any two points on the line, let's say point $A$ that has coordinates $(x_1, y_1)$ and point $B$ that has coordinates $(x_2, y_2)$ and applying the following formula:

(1)
\begin{align} m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{x_1 - x_2}{y_1 - y_2} \end{align}

For example, if we wanted to calculate the slope of a line that passes through the points $(2, 3)$ and $(-3, -1)$, applying the formula, we obtain that:

(2)
\begin{align} m = \frac{-1 - 3}{-3 - 2} = \frac{-4}{-5} = \frac{4}{5} \end{align}

We should note that the general order of the points $A$ and $B$ don't matter as both points still lie on the line.

## Slope-Intercept Form

One way to represent the equation of a line is in it's slope-intercept form $y = mx + b$ where $m$ represents the slope of the line and $b$ is the $y$-coordinate of the $y$-intersection.

For example, the equation $y = 2x + 3$ has a slope of $2$ and has a $y$-intersection at $(0, 3)$ as illustrated:

### Converting to Slope-Intercept Form

Sometimes it may be necessary to convert a line to it's slope-intercept form. We note that the slope-intercept form of a line is equivalent to the original equation of the line. To convert to slope-intercept form given a line in the form $ax + by + c = 0$, all we have to do is isolate $y$.

For example, consider the line $2x + 4y + 5 = 0$. The following procedure shows the isolation of $y$:

(3)
\begin{align} 2x + 4y + 5 = 0 \\ 4y = -2x - 5 \\ y = \frac{-2}{4}x - \frac{5}{4} \\ y = \frac{-1}{2}x - \frac{5}{4} \end{align}

Thus, the line $2x + 4y + 5 = 0$ has a slope of $\frac{-1}{2}$ and a $y$-intercept at $(0, \frac{5}{4})$.

## General Form of a Line

A line is said to be in general form if it is written in a manner such as $ax + by + c = 0$. We've already looked at a general form of a line in the last example, that was $2x + 4y + 5 = 0$. We will note that this form is important for finding the coordinates of an $x$-intercept later.

### Converting to General Form

Suppose that we have a line in the form $y = mx + b$. We can convert this line to general form by bringing all variables and numbers to one side of the equals sign and then eliminating any denominators by multiplication.

For example, the following math shows the procedure for converting the line $y = \frac{2}{3}x - 3$ into general form:

(4)
\begin{align} y = \frac{2}{3}x - 3 \\ 0 = \frac{2}{3}x - y - 3 \\ \: 0 \cdot 3 = \left ( \frac{2}{3}x - y - 3 \right) \cdot 3 \\ 0 = 2x - 3y - 9 \end{align}

Note that any multiple our general form is still the same line, for example, $0 = -2x + 3y + 9$ is the same line and we would have resulted in getting this as our answer if we brought all of our terms to the other side of the equals sign. Furthermore, $0 = -4x + 6y + 18$ is also the same line, though we generally try to divide by any common divisor of the terms for simplification.

## Example Questions

• 1. Given the line $y = 3x - 3$, what is the slope of the line and the coordinates of the $y$-intercept?
• 2. A line has a slope of $-2$ and a $y$-intercept at $(0, 6)$. What is the slope-intercept form of this line?
• 3. Convert $2x - 3y -1 = 0$ into slope-intercept form.
• 4. Convert $-4x - 11y + 2 = 4$ into slope-intercept form.
• 5. Convert $y = \frac{1}{3}x - 5$ into general form.
• 6. Convert $y = \frac{2}{3}x - \frac{5}{4}$ into general form.
• 7. A line has a slope of $-\frac{2}{3}$ and a $y$-intercept at $(0, 4)$. What is the general form of this line?