Main content

### Course: Algebra (all content) > Unit 3

Lesson 2: x-intercepts and y-intercepts- Intro to intercepts
- x-intercept of a line
- Intercepts from a graph
- Intercepts from an equation
- Intercepts from an equation
- Intercepts from a table
- Intercepts from a table
- Graphing using intercepts (old)
- Intercepts of lines review (x-intercepts and y-intercepts)

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Graphing using intercepts (old)

An old video of Sal where he draws the line y=3x-9 by finding its x- and y-intercetps. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- Do we always plug in zero for the variable? I have been having a lot of trouble with these questions and I watched all these videos, and every time Sal plugs in 0 for the variable and proceeds through the equation. Is this something that happens in every problem?(8 votes)
- When you are talking about the x intercept, where it crosses the x axis, you do not have to go up or down to get to that point; so your y value is zero.

When you are talking about the y intercept, where it crosses the y axis, you do not have to go left or right; so your x value is zero.

When you plug in a value for one variable you can use the equation to solve for the other, this creates a point that makes the equation true. which means it is a point on the line.

You can plug any value in for x or y with LINEAR equations (other types of equations will have domain and range restrictions), but the easiest value to plug in almost all cases is zero. Most students, when asked to graph lines in two variables, automatically start by drawing the x and y axis. Since your Intercepts lie on these axis, plotting the x and y intercepts comes naturally.

Other points can be found by plugging in various numbers for x and then solving for y, like you would in a table of values approach to graphing. With linear equations it only takes two points to graph the line. Or a point (preferably the y intercept) and the slope (which generates more points -so you have at least two).(2 votes)

- what would be an equation that passes through the point (3, 2) and creates a system of equations with 10x + 5y = 15 that has no solutions.(3 votes)
- Well, we know that for a system of linear equations to have 0 solutions, their graphs must be parallel. In other words, they have the same slope.

If we change 10x +5y = 15 into slope-intercept form, we get y = -2x +3. So the slope is -2, and the slope of the line we must find is also -2.

Given the point (3,2), we can use Point-Slope form to write the equation: y-2=-2(x-3). Written in slope intercept form, this is y = -2x +8, and written in standard form, it is 2x + y = 8.(4 votes)

- so when finding coordinates do we use the same strategy to plot on the graph

for example y= -1/3x+8 how would its graph look(2 votes)- I can use the b in your equation (y=mx+b) to know the y-intercept right off the bat. Then from there I know that m= rise/run. So I know that the next point will be 1 unit down and 3 units to the right. This graph will eventually be a line going down from left to right with a smooth slope.(3 votes)

- Ok, My questions is related to this.........If a graphed line is on the y axis and meets x at points (0,0) or a vertical line basically running on the y axis. What would the linear inequality equation be......? I assumed its y=x.....even my teacher does not know how!(2 votes)
- William,

The linear equation would be x=0

It could also be written as x=0y or x=0y+0

No matter what number you put in for y, x is always 0 which gives you a line on the y axis.

Note that this cannot be written as a function of x because one x value results in more than one answer for y (In fact you have an infinite number of y value when x is 0)

Also note that if you take two points on the line and find the slope as (change in y)/(change in x) you get a 0 for the denominator. Any fraction with a 0 in the denominator is undefined. So the slope of the line x=0 is undefined. Any vertical line will have an undefined slope.

I hope that helps make it click for you.(2 votes)

- At4:27, i need to practice!! I don't need a video!(1 vote)
- Becky,

Here is a practice session that is as close to the concept in this video as I have found. https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/graphing-slope-intercept/e/graphing_linear_equations

It might help!(4 votes)

- Using information from a graph how would you find a graph using slop and y-intercept?(2 votes)
- Find where the line cross the Y axis. Then find all the point on the graph and put the difference of y over the difference of x. That is your slope.(1 vote)

- If the x-intercept means that y=0, does that mean that the y-intercept would mean x=0? Is it always like that?(2 votes)
- Yes always.

For a straight line y = mx + b, for every number on the x axis the equation of the line gives you the y value (mx+b) on the line. the y you get for x=0 is called the "y intercept" (m(0) + b).

If you think of y as an independent variable, you can see that the same line "y=mx+b" has a different slope in relation to the y axis (solve "y = mx + b" for x to get "x = (1/m)y - b/m" to get this slope.). Now, for every value of y there is a value of x, = (1/m)y - b/m, on the line, and the x you get for y = 0, = (1/m)*0 - b/m, is the x-intercept.

But you don't have to go through all this - you can just use "y = mx + b", set y=0 and solve "mx+b = 0", which does the same thing..(2 votes)

- x-intercept of 4 and a y-intercept of 3, find the slope of a parallel line and the slope of a perpendicular line(2 votes)
- Parallel Line Slope: -3/4

Perpendicular Line Slope: 4/3

The Parallel Line's slope is equivalent to the slope of the given line, and the perpendicular line's slope is the negative reciprocal of the given line's slope, i.e. if the given line's slope is "m", then the perpendicular line's slope is "-1/m".(1 vote)

- for a function equation (y=mx + b)

why do we have to find the y intercept and not the x intercept.(2 votes)- y=mx+b is known as the "Slope-Intercept" form. Where m is the slope and b is the y-intercept (if x = 0, then y=b so the line intercepts the y-axis at b).(1 vote)

- So if the line never hits the x or y axis then is there just no intercept?(2 votes)
- Assuming it is coplanar, it will always intercept at least one axis, but if it is parallel to one of the axes, it will only have one intercept, at least in regard to the axis with which it is perpendicular. For example, if the line is parallel to the x axis, it will only have a y intercept, and vice versa.(1 vote)

## Video transcript

Identify the x and
y-intercepts of the line y is equal to 3x minus 9. Then graph the line. So the x-intercept,
I'll just abbreviate it as x-int, that is where the
line intersects the x-axis. So where it
intersects the x-axis. Remember, this horizontal
axis is the x-axis. So when something
intersects the x-axis, what do we know
about its coordinate? Its x-value could be anything,
but we know it's y-value is 0. If we're intersecting,
if we're sitting on the x-axis
someplace, that means that we haven't moved
in the y-direction. That means that y is 0. So this means,
literally, that y is 0. So we need to find
the x-value defined by this relationship
when y is equal to 0. Similarly, when we talk
about the y-intercept, I'll do it down
here-- when we talk about the y-intercept,
what does that mean? Well, y-intercept
means-- so this is the y-axis right over
here running up and down. The y-intercept is
the point at which the line intercepts the y-axis. So what's going on? If we're at the
y-axis, our y-value could be anything depending on
where we intersect the y-axis. But we know that we haven't
moved to the right or the left. We know that our x-value
is 0 at the y-intercept. So over here, our
x-value is going to be 0. And to find the
actual point, we just have to find the
corresponding y-value defined by this relationship
or this equation. So let's do the first one first. The x-intercept is
when y is equal to 0. So we set y is equal to 0,
and then we'll solve for x. So we get 0 is
equal to 3x minus 9. We can add 9 to both
sides of this equation to isolate the x-term. So we get 9 is equal to 3x. These cancel out. We could divide both sides by 3. Divide both sides by 3. We get 3 is equal to
x or x is equal to 3. So the point y is equal to 0, x
is equal to 3 is on this line. And let me put it in order. x-coordinate always goes first. So it's 3 comma 0. So this is the origin. 1, 2, 3 is right over here. That is 3 comma 0. This right here is
the x-intercept. And remember, notice that
point lies on the x-axis, but the y-value is 0. We haven't moved up or down. When you think
x-intercept, you say, OK, that means my y-value is 0. So I have to solve
for the x-value. Now we do the opposite
for the y-intercept. And the y-intercept, we're
sitting on this line, x-value must be 0. So let's figure out what y is
equal to when x is equal to 0. So y is equal to-- I want
to do it in that pink color. y is equal to-- y is equal
to 3 times-- x is 0 now. 3 times 0 minus 9. Well, 3 times 0 is just 0. So 0 minus 9. Well that's, just
equal to negative 9. So we have the point
0 comma negative 9. So when x is 0, we
go down 9 for y. 1, 2, 3, 4, 5, 6, 7, 8, 9. So right there is the
point 0 comma negative 9. Notice, it sits on the y-axis. That's why it's the y-intercept. And the x-value is 0. We haven't moved to
the left or right. All you need is two
points for a line, so we're now ready to graph. We essentially just have
to connect the dots. So it's going to look
something like this. It's going to look
something-- our line. I don't have a good
line tool, so I'm going to try my best
to draw it nicely-- is going to look
something like that. And you just keep going. You just keep going. You want to do a straight line. So it just keeps going on
and on and on like that. It just keeps going. And I could keep going all
the way in that direction, and then-- but then my line
doesn't look as straight all of a sudden. I think you get
the general idea. I can keep going like that,
and then keep going like that. I don't have a nice
ruler to do it with. And we're done.