Write an equation in slope-intercept form of the line that passes through (−5, 19) and (5, 13). y=

The slope of a line is its vertical change divided by its horizontal change, also known as rise over run. When you have 2 points on a line on a graph the slope is the change in y divided by the change in x.

The slope of a line is a measure of how steep it is.

Slope Calculator Solutions

Input two points using numbers, fractions, mixed numbers or decimals. The slope calculator shows the work and gives these slope solutions:

• Slope m with two points
• Graph of the line for y = mx + b
• Point Slope Form y - y1 = m(x - x1)
• Slope Intercept Form y = mx + b
• Standard Form Ax + By = C
• y-intercept, when x = 0
• x-intercept, when y = 0

You will also be provided with a custom link to the Midpoint Calculator that will solve and show the work to find the midpoint and distance for your given two points.

How to Calculate Slope of a Line

Calculate slope, m using the formula for slope:

Slope Formula

Here you need to know the coordinates of 2 points on a line, (x1, y1) and (x2, y2).

How to Find Slope of a Line

1. Find the difference between the y coordinates, Δy is change in y
2. Find the difference between the x coordinates, Δx is change in x
3. Divide Δy by Δx to find slope

Example: Find the Slope

Say you know two points on a line and their coordinates are (2, 5) and (9, 19). Find slope by finding the difference in the y points, and divide that by the difference in the x points.

1. The difference between y coordinates Δy is
2. The difference between x coordinates Δx is
3. Divide Δy by Δx to find slope m

\( m = \dfrac {14} {2} \)

Line Equations with Slope

There are 3 common ways to write line equations with slope:

• Point slope form
• Slope intercept form
• Standard form

Point slope form is written as

Using the coordinates of one of the points on the line, insert the values in the x1 and y1 spots to get an equation of a line in point slope form.

Lets use a point from the original example above (2, 5), and the slope which we calculated as 7. Put those values in the point slope format to get an equation of that line in point slope form:

If you simplify the point slope equation above you get the equation of the line in slope intercept form.

Slope intercept form is written as

Take the point slope form equation and multiply out 7 times x and 7 times 2.

Continue to work the equation so that y is on one side of the equals sign and everything else is on the other side.

Add 5 to both sides of the equation to get the equation in slope intercept form:

Standard form of the equation for a line is written as

You may also see standard form written as Ax + By + C = 0 in some references.

Use either the point slope form or slope intercept form equation and work out the math to rearrange the equation into standard form. Note that the equation should not include fractions or decimals, and the x coefficient should only be positive.

Slope intercept form: y = 7x - 9

Subtract y from both sides of the equation to get 7x - y - 9 = 0

Add 9 to both sides of the equation to get 7x - y = 9

Slope intercept form y = 7x - 9 becomes 7x - y = 9 written in standard form.

Find Slope From an Equation

If you have the equation for a line you can put it into slope intercept form. The coefficient of x will be the slope.

Example

You have the equation of a line, 6x - 2y = 12, and you need to find the slope.

Your goal is to get the equation into slope intercept format y = mx + b

1. Start with your equation 6x - 2y = 12
2. Add 2y to both sides to get 6x = 12 + 2y
3. Subtract 12 from both sides of the equation to get 6x - 12 = 2y
4. You want to get y by itself on one side of the equation, so you need to divide both sides by 2 to get y = 3x - 6
5. This is slope intercept form, y = 3x - 6. Slope is the coefficient of x so in this case slope = 3

How to Find the y-Intercept

The y-intercept of a line is the value of y when x=0.  It is the point where the line crosses the y axis.

Using the equation y = 3x - 6, set x=0 to find the y-intercept.

How to Find the x-Intercept

The x-intercept of a line is the value of x when y=0.  It is the point where the line crosses the x axis.

Using the equation y = 3x - 6, set y=0 to find the x-intercept.

Slope of Parallel Lines

If you know the slope of a line, any line parallel to it will have the same slope and these lines will never intersect.

Slope of Perpendicular Lines

If you know the slope of a line, any line perpendicular to it will have a slope equal to the negative inverse of the known slope.

Perpendicular means the lines form a 90° angle when they intersect.

Say you have a line with a slope of -4. What is the slope of the line perpendicular to it?

• First, take the negative of the slope of your line
  -(-4) = 4
• Second, take the inverse of that number. 4 is a whole number so its denominator is 1. The inverse of 4/1 is 1/4.
• The negative inverse of a slope of -4 is a slope of 1/4.
• A line perpendicular to your original line has a slope of 1/4.

Further Study

Brian McLogan (2014) Determining the slope between two points as fractions, 10 June. At https://www.youtube.com/watch?v=Hz_eapwVcrM

The slope calculator determines the slope or gradient between two points in the Cartesian coordinate system. The slope is basically the amount of slant a line has and can have a positive, negative, zero, or undefined value. Before using the calculator, it is probably worth learning how to find the slope using the slope formula. To find the equation of a line for any given two points that this line passes through, use our slope intercept form calculator.

1. Identify the coordinates (x₁,y₁)and(x₂,y₂). We will use the formula to calculate the slope of the line passing through the points (3,8) and (-2, 10).
2. Input the values into the formula. This gives us (10 - 8)/(-2 - 3).
3. Subtract the values in parentheses to get 2/(-5).
4. Simplify the fraction to get the slope of -2/5.
5. Check your result using the slope calculator.

To find the slope of a line we need two coordinates on the line. Any two coordinates will suffice. We are basically measuring the amount of change of the y-coordinate, often known as the rise, divided by the change of the x-coordinate, known the the run. The calculations in finding the slope are simple and involves nothing more than basic subtraction and division.

slope = (y₂ - y₁) / (x₂ - x₁)

Notice that the slope of a line is easily calculated by hand using small, whole number coordinates. The formula becomes increasingly useful as the coordinates take on larger values or decimal values.

It is worth mentioning that any horizontal line has a gradient of zero because a horizontal line has the same y-coordinates. This will result in a zero in the numerator of the slope formula. On the other hand, a vertical line will have an undefined slope since the x-coordinates will always be the same. This will result the division by zero error when using the formula.

Just as slope can be calculated using the endpoints of a segment, the midpoint can also be calculated. The midpoint is an important concept in geometry, particularly when inscribing a polygon inside another polygon with the its vertices touching the midpoint of the sides of the larger polygon. This can be obtained using the midpoint calculator or by simply taking the average of each x-coordinates and the average of the y-coordinates to form a new coordinate.

The slopes of lines are important in determining whether or not a triangle is a right triangle. If any two sides of a triangle have slopes that multiply to equal -1, then the triangle is a right triangle. The computations for this can be done by hand or by using the right triangle calculator. You can also use the distance calculator to compute which side of a triangle is the longest, which helps determine which sides must form a right angle if the triangle is right.

The sign in front of the gradient provided by the slope calculator indicates whether the line is increasing, decreasing, constant or undefined. If the graph of the line moves from lower left to upper right it is increasing and is therefore positive. If it decreases when moving from the upper left to lower right, then the gradient is negative.

The method for finding the slope from an equation will vary depending on the form of the equation in front of you. If the form of the equation is y=mx+c, then the slope (or gradient) is just m. If the equation is not in this form, try to rearrange the equation. To find the gradient of other polynomials, you will need to differentiate the function with respect to x.

1. Use a map to determine the distance between the top and bottom of the hill as the crow flies.
2. Using the same map, or GPS, find the altitude between the top and bottom of the hill. Make sure that the points you measure from are the same as step 1.
3. Convert both measurements into the same units. Use the Omni Length Converter if you are unsure.
4. Divide the difference in altitude by the distance between the two points.
5. This number is the gradient of the hill if it increases linearly. If it does not, repeat the steps but at where there is a noticeable change in slope.

1. Measure the difference between the top and bottom of the slope in relation to both the x and y axis.
2. If you can only measure the change in x, multiply this value by the gradient to find the change in the y axis.
3. Make sure the units for both values are the same.
4. Use Pythagoras’ theorem to find the length of the slope. Square both the change in x and the change in y.
5. Add the two values together.
6. Find the square root of the summation.
7. This new value is the length of the slope.

A 1/20 slope is one that rises by 1 unit for every 20 units traversed horizontally. So, for example, a ramp that was 200 ft long and 10 ft tall would have a 1/20 slope. A 1/20 slope is equivalent to a gradient of 1/20 (strangely enough) and forms an angle of 2.86° between itself and the x-axis.

As the slope of a curve changes at each point, you can find the slope of a curve by differentiating the equation with respect to x and, in the resulting equation, substituting x for the point at which you’d like to find the gradient.

The rate of change of a graph is also its slope, which are also the same as gradient. Rate of change can be found by dividing the change in the y (vertical) direction by the change in the x (horizontal) direction, if both numbers are in the same units, of course. Rate of change is particularly useful if you want to predict the future of previous value of something, as, by changing the x variable, the corresponding y value will be present (and vice versa).

Slopes (or gradients) have a number of uses in everyday life. There are some obvious physical examples - every hill has a slope, and the steeper the hill, the greater its gradient. This can be useful if you are looking at a map and want to find the best hill to cycle down. You also probably sleep under a slope, a roof that is. The slope of a roof will change depending on the style and where you live. But, more importantly, if you ever want to know how something changes with time, you will end up plotting a graph with a slope.

A 10% slope is one that rises by 1 unit for every 10 units travelled horizontally (10 %). For example, a roof with a 10% slope that is 20 m across will be 2 m high. This is the same as a gradient of 1/10, and an angle of 5.71° is formed between the line and the x-axis.

To find the area under a slope you need to integrate the equation and subtract the lower bound of the area from the upper bound. For linear equations:

1. Put the equation into the form y=mx+c.
2. Write a new line where you add 1 to the order of the x (e.g., x becomes x^2, x^2.5 becomes x^3.5).
3. Divide m by the new number of the order and put it in front of the new x.
4. Multiply c by x and add this to the new line.
5. Solve this new line twice, one where x is the upper bound of the area you wish to find and one where x is the lower bound.
6. Subtract the lower bound from the upper bound.
7. Congratulate yourself on your achievement.

A 5 to 1 slope is one that, for every increase of 5 units horizontally, rises by 1 unit. The number of degrees between a 5 to 1 slope and the x-axis is 11.3°. This can be found by first calculating the slope, by dividing the change in the y direction by the change in the x direction, and then finding the inverse tangent of the slope.