Positive Slope Explained: How It Looks and How to Find It

A positive slope characterizes a line that goes upward from left to right on a graph. This indicates a positive correlation between x and y, meaning that y-values increase as x-values increase. In this article, you will learn how to recognize a positive slope visually, as well as using linear equations and points belonging to a line. You will also see several examples to understand how the theory works in practice.

What is a positive slope? You already know that in visual terms, a positive slope represents a line that rises from left to right. But what does that mean from a mathematical standpoint?

Whenever a linear equation graph has a positive slope, you can identify it by examining the equation, particularly its slope-intercept form 🇺🇸.

$y = mx + b$

In this equation, $m$ is the slope of the line, while the y-intercept 🇺🇸 is given by (0, b).

A positive $m$ means that there is an increasing relationship between $x$ and $y$. The larger the value of $m$, the steeper the line.

Let's graph the following equation: $y=2x+1$.

This is an example of a positive slope, with $m=2$. As shown on the graph, the line rises two units for every unit it moves to the right. If $m$ were equal to $3$, it would increase three units, and so on. Regardless of the coefficient, the graph of a positive slope will always move upward from left to right. However, there are other ways of determining whether the slope is positive.

How to recognize a positive slope line from an equation

It's easy when the equation is in the slope-intercept form — you must identify the coefficient $m$. Take $y=\frac{3}{4}x - 3$. In this equation, $m=\frac{3}{4}$, hence, the slope is positive, because $m>0$.

But look at this other equation: $5x + 3y = 8$. Where is $m$?

It's impossible to identify $m$ from the standard form of a linear equation. You will first have to rewrite the equation, isolating $y$ on the left-hand side; only then can you identify $m$.

$-5x + 3y = 8$

$3y = 5x + 8$

$\large \frac{3y}{3} = \frac{5x+8}{3}$

$y = \frac{5}{3}x + \frac{8}{3}$

Do you see where $m$ is? Exactly, $m = \frac{5}{3}$. This is an example of a positive slope, because $m > 0$.

Now look at this other equation:

$\frac{1}{4}x + 2y = 10$

$2y = -\frac{1}{4}x + 10$

$\large \frac{2y}{2} = \frac{-\frac{1}{4}x + 10}{2}$

$y = -\frac{1}{8}x + 5$

We have identified that $m=-\frac{1}{8}$, but is this slope positive? No, this slope is negative, because $m<0$.

Identifying a positive slope line using two points

If you don't have access to the linear equation, but you can identify two points, $\text{A}(x_1,y_1)$ and $\text{B}(x_2,y_2)$, belonging to the line in question, you can calculate $m$ using their coordinates with the following equation:

$\large m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

This formula is called rise over run 🇺🇸. Once you have calculated $m$, you can determine whether the slope is positive.

Take $A(2,-6)$ and $B(5,7)$. Let's calculate the slope of the line passing through $A$ and $B$:

$\large m = \frac{\Delta y}{\Delta x} = \frac{7 - (-6)}{5 -2}$

$\large m = \frac{13}{3} = 4 \frac{1}{3}$

You know the drill: since $m>0$, the slope is positive. Look at the graph of this positive slope line to confirm:

As you can see, the line in this graph is steeper compared to the previous graph of $y=2x+1$, as $4\frac{1}{3} > 2$. Here, it rises $4\frac{1}{3}$ units for every unit it moves right.

Use our line equation from two points calculator 🇺🇸 to generate a linear equation passing through any two points.

Now you try! Take the following linear equations:

• $2x-y=5$
• $4x+y=3$

Are their slopes positive or negative? Remember the definition of a positive slope!

A positive slope is one of the fundamental concepts of coordinate geometry. Algebraically, you can identify it by looking at a linear equation or calculating it from two points belonging to a line. Visually, it's simply a line that rises from left to right on a graph. Whichever method you choose, we hope that recognizing your next positive slope will be a breeze!