Slope and Y-intercept Formula: How to Identify the Slope and Y-intercept

The slope and y-intercept formula of a linear equation, also known as the slope-intercept form 🇺🇸, is used in algebra to identify the y-intercept and the slope of a line. But what are the slope and y-intercept? Thanks to this article, you will discover this and more!

The slope and y-intercept are two crucial characteristics of a line:

• The slope represents the rate of change given by the line equation. Graphically, it shows how many units a line rises or falls for every unit it goes to the right.
• The y-intercept 🇺🇸 is the point of intersection between the line and the y-axis.

Both these elements are present in the slope and y-intercept formula:

$y=mx+b$

where:

• $m$ — Slope; and
• $b$ — Y-coordinate of the y-intercept, the x-coordinate being 0.

The slope and y-intercept formula makes finding the slope and y-intercept easy as pie — all you have to do is look at the equation and write down your observations. For example:

$y=5x-9$

In this equation:

• $m=5$
• $b=-9$

Therefore, the slope 🇺🇸 is equal to $5$, and the y-intercept is $Y(0,-9)$.

But what happens if the equation is written in a different form, or there is no equation at all?

Rearranging a linear equation

If your equation is written in a standard form, $Ax + By = C$, you can do two things:

• Rearrange the equation, such that $y$ is isolated on the left; or
• Use the following formulas to calculate the slope and the y-coordinate of the y-intercept:
  • $m=\frac{-A}{B}$
  • $b=\frac{C}{B}$

Take the following equation: $-4x+6y=9$.

Using our two formulas, we can perform the following calculations:

• $m=\frac{-A}{B}=\frac{-(-4)}{6}=\frac{4}{6}=\frac{2}{3}$
• $b=\frac{C}{B}=\frac{9}{6}=\frac{3}{2}$

Our slope and y-intercept formula would, therefore, be $y=\frac{2}{3}x + \frac{3}{2}$.

Let's confirm it by rearranging the equation:

$-4x+6y=9$

$6y=4x+9$

$y=\frac{4x+9}{6}$

$y=\frac{4}{6}x + \frac{9}{6}$

$y=\frac{2}{3}x + \frac{3}{2}$

Same as above!

Remember that if your equation is written in any other form, you cannot use the formulas for $m$ and $b$ that we've seen here — you will always have to rearrange the equation, isolating $y$ on one side, and one x-term and one constant on the other.

Graphing a line using the slope and y-intercept is an easy way to visualize our parameters without the need for complicated calculations. Here's how to graph the slope and y-intercept:

1. Plot the y-intercept, $Y(0,b)$.
2. The slope informs you of how many units the line rises or falls for every unit that it goes to the right. Therefore, count 1 unit to the right from $Y$, then count $m$ units up. This is your second point; let's call it $A$.
3. Draw a straight line passing through both points.

And there you have it — you have drawn your line without needing to calculate the coordinates of your points. Let's make a graph together to see how it works in practice!

Given $m=-3$ and $b=2$, plot a line on a graph.

Using our data, we can write an equation in the slope-intercept form: $y=-3x+2$. But we won't need that! Let's plot our y-intercept first ($Y(0,2)$):

Then, count 1 unit to the right and $3$ units down, as our slope is equal to $-3$. The coordinates of our new point are $A(1, -1)$:

Finally, draw the line:

How easy was that?

This method will only work with reasonably easy slopes and y-intercepts, as things get complicated when fractions and big numbers are involved. But you can definitely use it for your everyday plotting needs!

The slope and y-intercept formula makes our lives easy by providing two fundamental characteristics of a linear equation on a silver platter. We can obtain it by rearranging any other type of linear equation, such as the standard form. Thanks to this article, you now know how to find the slope and y-intercept in any situation, and you can easily plot a line using only these two parameters. There's nothing left to say apart from happy graphing!