What is the equation of the line that passes through the point and has a slope of calculator

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Slope Formula
Steps to Find Slope Using the Formula
How to Interpret Slope
How to Find Equations of a Line Using the Slope
How to Find Slope-Intercept Form
How to Find Point-Slope Form
How to Find Standard Form
How to Find the x and y Intercepts of a Line
How to Find a Point Given 1 Point, Slope, and Distance
How to Convert Slope to Angle
How to Convert Angle to Slope
How to Find the Distance Between 2 Points

If we have a point,		, and a slope, m, here's the formula we
use to find the equation of a line:

It's called the point-slope formula
(Duh!)

You are going to use this a LOT!

Luckily, it's pretty easy -- let's just do one:

Let's find the equation of the line that passes through the point
( 4 , -3 ) with a slope of -2:

Just stick the stuff in a clean it up!

TRY IT:

Find the equation of the line that passes through the point ( -2 , 6 ) with a slope of 3.

Use the slope calculator to find the slope of a line that passes through 2 points or solve a coordinate given m. Plus, the calculator will also solve the slope-intercept form of a line.

Slope is a measure of the rate of incline or steepness of a line on a graph. You can find the slope of a line by comparing any 2 points on the line. A point is an x and y value of a cartesian coordinate on a grid.

Slope, expressed m, is equal to the rise between two coordinates on a line over the run, or rather, it’s the ratio of rise to run.

Rise is the vertical change of the line, or the change in altitude or elevation, and the run is the horizontal change, or the change in distance.[1]

Slope Formula

You can find the slope m of a line using the following formula:[2]

m = y2 – y1x2 – x1

Thus, the slope m is equal to y2 minus y1, divided by x2 minus x1.

Steps to Find Slope Using the Formula

To find the slope of a line using this formula, you’ll need to find 2 points along a line and find their x and y values. The value of x is the horizontal distance of the point from the vertical y-axis, and y is the vertical distance of the point from the horizontal x-axis.

Then, follow three easy steps to calculate the slope.

Step One: Substitute Points in the Slope Formula

Start by replacing the y1, y2, x1, and x2 values in the formula with the coordinates for the two points on the line.

Step Two: Find the Difference in x and y Coordinates

Next, solve the numerator and denominator of the slope formula using these values. The result will be the absolute value of the distance between the x values or y values of two points.

The numerator represents the change in the value of y, which is known as the delta of y, or just Δy. You can find the delta of y using the formula:

Δy = y2 – y1

The denominator represents the change in the value of x, which is known as the delta of x, or Δx. You can find the delta of x using the formula:

Δx = x2 – x1

Step Three: Divide the Delta of y by the Delta of x

The final step is to divide the delta of y by the delta of x to find the slope. You can do this by dividing the numerator in the slope formula by the denominator, or more simply by using a fraction to decimal calculator.

In some cases, you may want to represent the slope as a fraction or ratio. In this case, you can simplify the fraction to reduce it to its simplest form rather than converting it to a decimal.

For example, let’s find the slope of a line that passes through the points (3,2) and (11,8).

Start with the slope formula:

m = (y2 – y1)(x2 – x1)

Replace the x and y values with the coordinate’s x and y values.

m = 8 – 211 – 3

Solve the numerator and denominator to find the delta of x and y.

m = 8 – 211 – 3 = 68

Then, reduce the fraction to its simplest form.

m = 68 = 34

So, the slope m is equal to 34

The slope formula is useful for points along a linear line, but when working with non-linear functions, you might need to use an average rate of change calculator.

How to Interpret Slope

While the slope of a line doesn’t tell us where the line is located on the graph, it does tell us the angle of the line.

If the slope is positive, then the line is slanted up and rises from left to right on the graph.

If the slope is negative, then the line is slanted down and drops from left to right on the graph.

If the slope is equal to zero, the line is a horizontal line and does not slant up or down. This is a line with no slope.

In the special case of a vertical line, the slope is undefined. Essentially the slope would be infinite in this case, and because this would result in the denominator of the slope formula being zero since there is no change in x, it is undefined.

How to Find Equations of a Line Using the Slope

Now that you found the slope, you can express a linear line using one of several equations. The most common equations for a line are slope-intercept form, point-slope form, and standard form.

How to Find Slope-Intercept Form

Slope-intercept form is one of the most commonly used equations to represent a linear line. You can find the equation of a line using slope-intercept form given the slope m and one point on the line.

Slope-Intercept Form Equation

The slope-intercept form equation is:[3]

y = mx + b

The slope-intercept form equation states that the y value of a coordinate on the line is equal to the x value of the coordinate times the slope m, plus the y-intercept b.

For example, let’s solve for b, given a slope of 1/2 and a point (5,4). We’ll express this line in slope-intercept form.

Substitute for m:

y = 12 × x + b

Substitute the point values for x and y:

4 = 12 × 5 + b

Solve for b:

4 = 2.5 + b
4 – 2.5 = b
1.5 = b

Substitute b in the equation to find the slope-intercept form:

y = 12x + 1.5

How to Find Point-Slope Form

Point-slope form is another commonly used equation to represent a line. You can find the equation of a line using point-slope form given the slope m and one point on the line.

Point-Slope Form Equation

The point-slope form equation is:[4]

y – y1 = m(x – x1)

The point-slope form equation states that y minus the y1 coordinate is equal to the slope m times x minus the x1 coordinate.

For example, let’s express the same line from the previous example using point-slope form. Recall the line has a slope of 1/2 and a point (5,4).

Substitute for m:

y – y1 = 12(x – x1)

Substitute the point values for x and y:

y – 4 = 12(x – 5)

So, the point-slope form for this line is y – 4 = 12(x – 5).

How to Find Standard Form

Standard form is a standardized format for linear equations. To use standard form, the slope and coordinates must be whole numbers; decimal values are not supported in standard form.

Standard Form Equation

The standard form equation is:

Ax + By = C

Given either of the line equations above, you can convert to standard form by simply rearranging the equations to isolate C on one half of the equation. If there are any fractions, then multiply both sides of the equation by the denominator of the fractions to remove them.

This formula represents the standard form for linear equations, but numbers can also be expressed in standard form as well.

How to Find the x and y Intercepts of a Line

Given the slope of a line and one point on it, you can find the x and y intercepts using one of the line equations above. Start by finding the slope-intercept form equation for the line.

Then, given slope-intercept form, you can find the y-intercept by setting x in the equation to 0 and solving.

And to find the x-intercept, set y in the equation to 0 and solve it using algebra.

For example, let’s find the x and y intercepts of the line y = 12x + 8.

Start by setting x to 0.

y = (12 × 0) + 8

Then solve for y.

y = 8

When x is equal to zero y is equal to 8, so 8 is the y-intercept.

Next, set y to 0.

0 = 12x + 8

Then solve for x.

0 = 12x + 8
0 – 8 = 12x + 8 – 8
-8 = 12x
-81 = 12x
-8 × 21 = 1 × 22x
-161 = 22x
-16 = x

When y is equal to zero, x is equal to -16, so the x-intercept is -16.

How to Find a Point Given 1 Point, Slope, and Distance

Points on a line can be solved given the slope of the line and the distance from another point. The formulas to find x and y of the point to the right of the point are:

x2 = x1 + d(1 + m2)

y2 = y1 + m × d(1 + m2)

The formulas to find x and y of the point to the left of the point are:

x2 = x1 + -d(1 + m2)

y2 = y1 + m × -d(1 + m2)