## What Is a Negative Slope?

**Negative slope refers to the slope of a line that is trending downwards as we move from left to right. **In mathematics, the slope of a line is the change in y-coordinate with respect to the change in x-coordinate.

So, **what does negative slope mean?** A negative slope means that two variables are negatively related. When x increases, y decreases.

**A line that makes an obtuse angle (an angle greater than 90 degrees) with the positive x-axis in the counterclockwise direction is known as a line with a negative slope. **If the line makes an acute angle with the positive direction of the x-axis, the line has a positive slope. A simple real-life example of negative slope is going down a hill.

## Negative Slope: Negative Rise Over Run Ratio

We know that the slope is given by the rise over run ratio.

Slope $= m = \frac{Rise}{Run} = \frac{\Delta y}{\Delta x}$

We can also say that a line has a negative slope when it has a negative rise over run ratio. Rise is the change in y-coordinates, and run is the change in x-coordinates. Let’s take a look at example of the negative slope equation with graphs showing rise over run ratio.

**Example 1:** Here, the line AB has a slope of $\frac{-1}{2}$.

Equation of line is $y = \frac{-1}{2}x + 1$

Slope $= \frac{Rise}{Run} = \frac{-1}{2}$

**Example 2:** Here, the line $y = \frac{-1}{3}x + 5$ has the rise over run ratio equal to $\frac{-1}{3}$.

## Negative Slope Graph

A line with a negative slope goes downward as we move in the positive direction of the x-axis. Mathematically, it means that as x increases, y decreases.

So, what does a negative slope look like? Take a look at the blue line shown below.

You can see that the blue line is trending downwards as it moves from left to right.

## How to Calculate Negative Slope

Let’s discuss methods to calculate negative slope.

**Method (i)**

When the coordinates of two points on the line are given, we can use this method.

**Example:** A line passes through points (4,2) and (3,5). To find if the line has a negative slope, we can use the slope formula.

Let us assume $(x_1,\;y_1) = (4,\;2)$ and $(x_2,\;y_2) = (3,\;5)$

Slope(m) $= \frac{y_2\;-\;y_1}{x_2\;-\;x_1}$

Then we have m $= \frac{3 – 4}{5 – 2} = \frac{-1}{3}$

Since m has a negative value, the line passing through points (4,2) and (3,5) has a negative slope.

**Method (ii)**

If the equation of a line is of the form $ax + by = c$, then we can use this method.

**Example:** The equation of a line is $2x + 3y = 5$

We can rewrite the equation as $y= \frac{-2}{3}x + 5$

If we equate the above equation with the general equation of the line in the slope-intercept form which is $y = mx + c$ (here in the equation m is the slope of the line and c is the intercept made by the line with the x-axis)

Equating the equation $y = \frac{-2}{3}x + 5$ with $y = m x + c$

We get $m = \frac{-2}{3}$

Since m has a negative value, the line $2x + 3y = 5$ has a negative slope.

**Method (iii)**

If the angle made by the line with the x-axis is given, then we can use this method.

**Example: **If a line makes an angle of 135 degrees with the positive direction of the x-axis then to find the negative slope, we use the slope formula.

Since the line makes an angle of 135 degrees with the positive direction of the x-axis, the slope of that line will be

$m = tan\; 135^\circ = tan\; (90 + 45) = -\;tan\; (45)= \;-1$

Thus, the given line has a negative slope.

## Types of Slope

There are four types of slope.

**Positive slope**: A line that makes an acute angle with the positive direction of the x-axis has a positive slope. The line with a positive slope rises up as we move from left to right.**Negative Slope**: A line that makes an obtuse angle with the positive x-axis has a negative slope. The line with a negative slope sinks down as we move from left to right.**Zero slopes**: A line that makes an angle of 0 degrees with the positive x-axis has zero slope. The line with zero slope does not have a rise, it remains horizontal, parallel to the x-axis.**Undefined slope:**A line that makes 90 degrees with the positive x-axis has an undefined slope. The line with an undefined slope is a vertical line or a line parallel to the y-axis.

## Facts about Negative Slope

- There are four types of slopes: (i) positive slope (ii) negative slope (iii) zero slope and (iv) undefined slope
- The negative slope makes an angle greater than 90 degrees with the positive direction but makes an angle less than 90 degrees with the negative direction of the x-axis.

## Conclusion

In this article, we learned about the negative slope and properties of lines having a negative slope. Let’s solve a few solved examples and practice problems based on these concepts.

## Solved Examples on Negative Slope

**1. Find whether the line passing through the points (5,2) and (2,-5) has a negative slope.**

**Solution: **

We have the slope formula

Slope$(m) = \frac{y_2\;-\;y_1}{x_2\;-\;x_1}$

Let us assume $(x_1,\;y_1) = (5,\;2)$ and $(x_2,\;y_2) = (2,\;-5)$

Now, we have to find $y_2\;-\;y_1 = (\;-5)\;-2 = \;-7$

Again, we are to find $x_2\;-\;x_1 = 2\;-\;5 = \;-3$

Now using the formula, Slope$(m) = \frac{y_2\;-\;y_1}{x_2\;-\;x_1} =\frac{-7}{-3} = \frac{7}{3}$

Since the value of the slope(m) is positive, the line passing through the points (5,2) and (2,-5) does not have a negative slope.

**2. Show that the line with equation **$5x + 2y = 5$** has a negative slope.**

**Solution: **

Given equation of a line is** **$5x + 3y = 5$.

Rewriting the equation as $y = \frac{-5}{3} x + 5$ to compare it with the general form of a line in slope-intercept form, which is $y = m x + c$.

Hence, we get the value of $m = \frac{-5}{3}$ which is a negative number.

So, the line ** **$5x + 3y = 5$ has a negative slope.

**3. Show that the line that makes an angle of 60 degrees with the positive direction of the x-axis does not have a negative slope.**

**Solution: **

We have a formula of slope(m) $= tan\; \theta$; here$\theta$ is the angle made by the line with the positive direction of the x-axis.

It is given that the line makes an angle of 60 degrees with the positive direction of the x-axis.

Hence, slope(m) $= \tan\;60^\circ = \sqrt{3}$, which is a positive value.

Therefore, the line with an angle of 60 degrees along the positive direction of the x-axis does not have a negative slope.

**4. Show that the line that makes an angle of 150 degrees with the positive direction of the x-axis has a negative slope.**

**Solution: **We have a formula of slope(m) $= tan\; \theta$; here $\theta$ is the angle made by the line with the positive direction of the x-axis.

It is given in the question that the line makes an angle of 150 degrees with the positive direction of the x-axis.

Hence, slope$(m) = tan\; 150^\circ = tan\;(90 + 60) =\;-\frac{1}{3}$, which is a negative value.

Therefore the line with an angle of 150 degrees along the positive direction of the x-axis has a negative slope.

## Practice Problems on Negative Slope

## Negative Slope: Definition, Graph, Solved Examples, Facts

### The line that makes an angle greater than 90 degrees with the positive x-axis has __________ slope.

The line that makes an angle greater than 90 degrees along the direction of the positive x-axis has a negative slope.

### If a line makes an angle of 45 degrees with the negative direction of the x-axis then the line has ______________.

If a line makes an angle of 45 degrees with the negative direction of the x-axis, then that line makes an angle of $(180\;-\;45) = 135$ degrees with the positive direction of the x-axis, which is an obtuse angle. So, the line has a negative slope.

### If a line has an equation $3x\;-\;3y = 1$, then the line has a _________ slope.

The equation of a line is $3x\;-\;3y = 1$.

Rewrite it as $y = x \;-\; \frac{1}{3}$

Comparing it with the equation $y = mx + c$, we get $m = 1$, which means that the line has a positive slope.

### If a line has an equation $3x + 3y = 8$ then this line has a ____________ slope.

The equation of a line is $3x + 3y = 8$

Rewrite it as $y = \;-x + \frac{8}{3}$

Comparing it with the equation $y = mx + c$, we get $m = -1$, which means that the line has a negative slope.

## Frequently Asked Questions on Negative Slope

**How to find the slope using rise over run?**

Rise over run is just a non-technical term for defining slope.

Slope $= \frac{Rise}{Run} = \frac{\Delta y}{\Delta x} = \frac{y_2\;-\;y_1}{x_2\;-\;x_1}$

**What is the rise and run slope?**

The slope of a line that is expressed in a fraction is referred to as rise over run. The numerator is the rise, which describes the change in y and the denominator of that fraction is run, which describes the change in x.

**How do you know if a slope is negative or positive using the graph?**

If the graph of a line rises from left to right, the slope is positive. Here, as x increases, y increases.

If the graph of the line falls from left to right the slope is negative. Here, as x increases, y decreases. Here, as x increases, y decreases.

**Does a line with a negative slope look like a horizontal line or vertical line?**

A line with a negative slope does not look like a horizontal or vertical line. A negative slope line goes downward as we move along the x-axis.

**Can the slope be negative?**

Yes, the slope of a line is negative when the line falls as we move from left to right. Here, the rise over run ratio is negative.