Graph the line with slope passing through the point calculator

Video transcript

- [Instructor] We are told graph a line with the slope of negative two, that contains the point four comma negative three. And we have our little Khan Academy graphing widget right over here, where we just have to find two points on that line, and then that will graph the line for us. So pause this video and even if you don't have access to the widget right now, although it's all available on Khan Academy, at least think about how you would approach this. And if you have paper and pencil handy, I encourage you to try to graph this line on your own, before I work through it with this little widget. All right, now let's do it together. So we do know that it contains point four comma negative three. So that's I guess you could say the easy part, we just have to find the point x is four y is negative three. So it's from the origin four to the right, three down. But then we have to figure out where could another point be? Because if we can figure out another point, then we would have graphed the line. And the clue here is that they say a slope of negative two. So one way to think about it is, we can start at the point that we know is on the line, and a slope of negative two tells us that as x increases by one, y goes down by two. The change in why would be negative two. And so this could be another point on that line. So I could graph it like this is x goes up by one, as x goes from four to five, y will go, or y will change by negative two. So why we'll go from negative three to negative five. So this will be done, we have just graphed that line. Now another way that you could do it, because sometimes you might not have space on the paper, or on the widget to be able to go to the right for x to increase, is to go the other way. If you have a slope of negative two, another way to think about it is, if x goes down by one, if x goes down by one, then y goes up by two. 'Cause remember, slope is change of y over change in x. So you could either say you have a positive change in y of two when x has a negative one change, or you could think of it when x is a positive one change, y has a negative two change. But either way notice, you got the same line. Notice this line is the same thing, as if we did the first way is we had x going up by one and y going down by two, it's the exact same line.

Straight lines can be represented in the cartesian plane in the form of linear equations of a certain value of the slope. These are obtained by point-slope, two-point forms, and many other techniques depending on the given information.

Answer: The equation of the line with slope m = 7 and point (2, 9) is y = 7x - 5.

Let's understand the solution in detail.

Explanation:

Before plotting on the graph, we need to find the equation of the line.

We use the point-slope form to find the equation of the line: y - \((y)_{1}\) = m (x - \((x)_{1}\)).

Here m = 7 and (\((x)_{1}\), \((y)_{1}\)) = (2, 9).

⇒ y - 9 = 7 (x - 2)

⇒ y - 9 = 7x - 14

⇒ y = 7x - 5

Now, we plot it on the graph.

Check out the online slope calculator.

Hence, the equation of the line with slope m = 7 and point (2, 9) is y = 7x - 5.

Our online expert tutors can answer this problem

Get step-by-step solutions from expert tutors as fast as 15-30 minutes. Your first 5 questions are on us!

Number Line

Calculator Use

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

Δy = y2 - y1

2. Find the difference between the x coordinates, Δx is change in x

Δx = x2 - x1

3. Divide Δy by Δx to find slope

m = Δy/Δx

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

Δy = y2 - y1

Δy = 19 - 5

Δy = 14

2. The difference between x coordinates Δx is

Δx = x2 - x1

Δx = 9 - 2

Δx = 7

3. Divide Δy by Δx to find slope m

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

\(m = 7 \)

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

y - y1 = m(x - x1)

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:

y - 5 = 7(x - 2)

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

y = mx + b

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

y - 5 = 7(x - 2)

y - 5 = 7x - 14

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:

y = 7x - 9

Standard form of the equation for a line is written as

Ax + By = C

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.

y = 3(0) - 6

y = -6

The y-intercept is -6

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.

0 = 3x - 6

3x = 6

x = 2

The x-intercept is 2

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 the slope of the line that passes through the points (