Y-intercept refers to the point where the line cuts the y-axis. It is generally denoted using only the y-coordinate of the point (the x-coordinate will always be zero).

In this tutorial, you’ll learn about the equation of a line in the slope-intercept form. So let’s get going.

The slope-intercept form is one of the different forms in which an equation of a line can be written. And it’s a very popular form.

This form is most useful when you want to write the equation of a line when its slope and the y-intercept are known.

So for the example above, the y-intercept would be (0, b) or simply b.

If a line has a slope m and cuts the y-axis at (0, b), the equation of the line in the slope-intercept form is –

Here’s a visual representation of the idea.

Consider the figure below.

We have a line – with slope m – that cuts the y-axis at $\hspace{0.2em} (0, b) \hspace{0.2em}$. So its $\hspace{0.2em} y \hspace{0.2em}$-intercept is $\hspace{0.2em} b \hspace{0.2em}$. Also, $\hspace{0.2em} (x, y) \hspace{0.2em}$ is an arbitrary point on the line.

Now, let’s get an expression for the slope (m) of the line. We know that

$\text{slope} = \frac{\text{rise}}{\text{run}}$

Where “rise” is the vertical distance between two points (difference between y-coordinates) on the line and “run” is the horizontal distance (difference between x-coordinates) between them.

Substituting the values from the graph, we have

$\begin{align*} m &= \frac{y - b}{x - 0} \\[1.3em] m &= \frac{y - b}{x} \end{align*}$

Multiplying both sides by x and rearranging –

$y = mx + b$

And there we have the equation of the line in its slope-intercept form.

Example

Write the equation of a line that has a slope of $\hspace{0.2em} 2 \hspace{0.2em}$ and cuts the $\hspace{0.2em} y \hspace{0.2em}$-axis at $\hspace{0.2em} (0,5) \hspace{0.2em}$.

Solution

We know the equation of a line in its slope-intercept form is –

$y = {\color{Red} m} x + {\color{Teal} b}$

The question tells us that slope (m) is $\hspace{0.2em} 2 \hspace{0.2em}$ and the $\hspace{0.2em} y \hspace{0.2em}$-intercept $\hspace{0.2em} (b \hspace{0.2em}$) is $\hspace{0.2em} 5 \hspace{0.2em}$. Remember, for the $\hspace{0.2em} y \hspace{0.2em}$-intercept, we are concerned only with the $\hspace{0.2em} y \hspace{0.2em}$-coordinate of the point where the line intersects the $\hspace{0.2em} y \hspace{0.2em}$-axis.

So plugging the values into the equation above, we have

$y = {\color{Red} 2} x + {\color{Teal} 5}$

That’s it.

Example

Write the equation of the line shown in the figure.

Solution

The only way this example is different from the previous one is that here, we need to extract the information from the figure.

The figure shows that the slope $\hspace{0.2em} (m \hspace{0.2em}$) is $\hspace{0.2em} \frac{5}{7} \hspace{0.2em}$ and the y-intercept $\hspace{0.2em} (b \hspace{0.2em}$) is $\hspace{0.2em} 3 \hspace{0.2em}$. We just need to substitute these values into the standard equation.

$\begin{align*} y &= {\color{Red} m} x + {\color{Teal} b} \\[1.3em] y &= {\color{Red} \frac{5}{7}} x + {\color{Teal} 3} \end{align*}$

And that’s our equation.

Example

Rewrite the following straight-line equation in the slope-intercept form.

$4x - 3y = 2$

Solution

There are three steps to converting any straight line equation into the slope-intercept form.

Step 1. Make sure you have only the $\hspace{0.2em} y \hspace{0.2em}$-term is on the left side.

In the equation given to us, there’s an $\hspace{0.2em} x \hspace{0.2em}$-term on the left that needs to move.

$4x - 3y = 2$

Subtracting $\hspace{0.2em} 4x \hspace{0.2em}$ from both sides

$-3y = -4x + 2$

Step 2. Divide the whole equation by the coefficient of $\hspace{0.2em} y \hspace{0.2em}$ so that the $\hspace{0.2em} y \hspace{0.2em}$-coefficient becomes $\hspace{0.2em} 1 \hspace{0.2em}$.

The coefficient of $\hspace{0.2em} y \hspace{0.2em}$ is $\hspace{0.2em} -3 \hspace{0.2em}$, so we divide throughout by $\hspace{0.2em} -3 \hspace{0.2em}$.

$\begin{align*} \frac{-3y}{-3} = \frac{-4x}{-3} + \frac{2}{-3} \\[1.3em] y = \frac{4x}{3} - \frac{2}{3} \end{align*}$

Step 3. Separate the $\hspace{0.2em} x \hspace{0.2em}$-term and the constant term.

They are already separate in our case here. So nothing more to do. The last equation above is our answer.

Example

Rewrite the following straight-line equation in the slope-intercept form.

$y - 8 = 2(x + 1)$

Solution

Again, the same three steps.

Step 1. Make sure only the $\hspace{0.2em} y \hspace{0.2em}$-term is on the left side.

$y - 8 = 2(x + 1)$

Adding $\hspace{0.2em} 8 \hspace{0.2em}$ to both sides –

$y = 2(x + 1) + 8$

Step 2. Divide the whole equation by the coefficient of $\hspace{0.2em} y \hspace{0.2em}$ so that the y-coefficient becomes $\hspace{0.2em} 1 \hspace{0.2em}$.

The coefficient of $\hspace{0.2em} y \hspace{0.2em}$ is already $\hspace{0.2em} 1 \hspace{0.2em}$. So we can skip this step.

Step 3. Separate the $\hspace{0.2em} x \hspace{0.2em}$-term and the constant term.

The equation above has $\hspace{0.2em} x \hspace{0.2em}$ and $\hspace{0.2em} 1 \hspace{0.2em}$ (a constant term) grouped together inside parentheses. So we open the parentheses by distributing $\hspace{0.2em} 2 \hspace{0.2em}$ across the terms.

$\begin{align*} y = 2x + 2 + 8 \\[1em] y = 2x + 10 \end{align*}$

Done.

And that brings us to the end of this tutorial on the slope-intercept form equation of a line. Until next time.

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