What is an equation in point-slope form of the line that passes through (−7, 1) and (−3, 9)?

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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

\[ m = \dfrac {(y_{2} - y_{1})} {(x_{2} - x_{1})} \] \[ m = \dfrac{rise}{run} = \dfrac{\Delta y}{\Delta x} = \dfrac{y_2 - y_1}{x_2 - x_1} \]

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

What is an equation in point-slope form of the line that passes through (−7, 1) and (−3, 9)?

The equation is useful when we know:

  • one point on the line: (x1, y1)
  • and the slope of the line: m,

and want to find other points on the line.

Have a play with it (move the point, try different slopes):

Now let's discover more.

What does it stand for?

What is an equation in point-slope form of the line that passes through (−7, 1) and (−3, 9)?

(x1, y1) is a known point

m is the slope of the line

(x, y) is any other point on the line

It is based on the slope:

What is an equation in point-slope form of the line that passes through (−7, 1) and (−3, 9)?

Slope m  =   change in y change in x   =   y − y1 x − x1

Starting with the slope:

we rearrange it like this:

to get this:

 
What is an equation in point-slope form of the line that passes through (−7, 1) and (−3, 9)?

So, it is just the slope formula in a different way!

Now let us see how to use it.

What is an equation in point-slope form of the line that passes through (−7, 1) and (−3, 9)?

slope "m"  =  31  =  3

y − y1 = m(x − x1)

We know m, and also know that (x1, y1) = (3, 2), and so we have:

That is a perfectly good answer, but we can simplify it a little:

y − 2 = 3x − 9

y = 3x − 9 + 2

y = 3x − 7

What is an equation in point-slope form of the line that passes through (−7, 1) and (−3, 9)?

m = −3 1 = −3

y − y1 = m(x − x1)

We can pick any point for (x1, y1), so let's choose (0,0), and we have:

y − 0 = −3(x − 0)

Which can be simplified to:

What is an equation in point-slope form of the line that passes through (−7, 1) and (−3, 9)?

What is the equation for a vertical line?
The slope is undefined!

In fact, this is a special case, and we use a different equation, like this:

Every point on the line has x coordinate 1.5,
that’s why its equation is x = 1.5

What About y = mx + b ?

You may already be familiar with the y=mx+b form (called the slope-intercept form of the equation of a line).

It is the same equation, in a different form!

The "b" value (called the y-intercept) is where the line crosses the y-axis.

So point (x1, y1) is actually (0, b)

and the equation becomes:

Start withy − y1 = m(x − x1)

(x1, y1) is (0, b):y − b = m(x − 0)

Which is:y − b = mx

Put b on other side:y = mx + b

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