Main content

### Course: Equations & Inequalities 229-236 > Unit 3

Lesson 2: Systems of equations with graphing- Systems of equations with graphing
- Systems of equations with graphing
- Systems of equations with graphing: y=7/5x-5 & y=3/5x-1
- Systems of equations with graphing: 5x+3y=7 & 3x-2y=8
- Systems of equations with graphing: chores
- Systems of equations with graphing: exact & approximate solutions
- Systems of equations with graphing

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Systems of equations with graphing: 5x+3y=7 & 3x-2y=8

To solve a system of linear equations by graphing, you need to graph each equation separately. Find two points on each line and connect them. The point where the two lines intersect is the solution to the system of equations. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- Sal started with these two equations:
`5x + 3y = 7`

`3x - 2y = 8`

He didn't put them into point slope form, but played around with them and got some results. Is this called a certain form, and will this always work?(36 votes)- This is called Standard form, but it is not there in the video!(5 votes)

- What if the two lines end up not intersecting? Are you still "solving the system using the graphing method"?(7 votes)
- If the lines do not intersect at all, there is no solution. If they intercept only one, there is one solution. If there seems to only be one line, there are infinite (Neverending) solutions.(3 votes)

- What if you get a fraction for the y-intercept when putting the equation(s) into slope-intercept form (y=mx+b)?(5 votes)
- There's no problem if you get a fraction for the y-intercept. For example, let's say I have a line whose slope is 2 and y-intercept is 3/5. The slope-intercept form of this line would be: y = 2x + 3/5(6 votes)

- I'm still confused...Why did it transform into a fraction?(3 votes)
- Sal is solving these by finding the intercepts, intercepts always have a 0 somewhere. x intercepts have y=0 (x,0) where x is a number, and y intercepts have x=0 (0,y). Doing this, he ends up with 3y=7. To isolate the y, you have to divide both sides by 3 which ends up as a fraction y=7/3. Same thing happens with x intercept.(3 votes)

- what about problems that have something like y=3x?(3 votes)
- In such a situation, the y-intercept, and the x- intercept are both (0,0). You can tell this, because if you plug in 0 for either x or y, or get 0 for the other. EX:

y=3x → 0=3x → 0/3=x=0

In order to graph, you would draw a line extending from the origin, increasing at a rate of 3 units. Consider the video**Algebra: graphing lines 1**(3 votes)

- So you just graph the slope?(2 votes)
- Yes except carry the eight on the right.(4 votes)

- I'll give an example.

y=6x+3

1.)The 6x is really saying its 6/1 aka...(rise over run)

2.)3 is the y-axis, so start at 3 on the y-axis.remember 3.)that its positive.

4.)Then 6/1= move 1 in the x-axis,then 6 in the y-axis,and you'll get your line.

How to convert this 4x-4y=8 into y=x-2

Lets start by moving the 4x part onto the other side of the equal sign,and we can do that by subtracting 4x from both sides.

1.)4x-4y=8

2.)-4y=8-4x

3.)y=-2+1x

4.)y=x-2(3 votes) - why why why why why why why why(3 votes)
- Why what? I get that you're currently suffering from doing Khan Academy, but you have to keep going. There's going to be a light at the end of the tunnel, you just have to make it there.(1 vote)

- At the end of the 2000 WNBA regular season the Houston comets had 22 more victories than loss is the number of victories they had were three less than six times the number of losses how many regular season games did the Houston comets play during the 2000 WNBA season?(2 votes)
- We do all that just to subtract 10-7?(2 votes)

## Video transcript

Solve the system of linear
equations by graphing, and they give us
two equations here. 5x plus 3y is equal to 7, and
3x minus 2y is equal to 8. When they say, "Solve the
system of linear equations," they're really just
saying find an x and a y that satisfies
both of these equations. And when they say to
do it by graphing, we're essentially going to
graph this first equation. Remember, the graph is
really just depicting all of the x's and y's that
satisfy this first equation, and then we graph
the second equation that's depicting all of the x's
and y's that satisfy that one. So if we were looking for an
x and a y that satisfies both, that point needs to
be on both equations or it has to be on both graphs. So it'll be the intersection
of the two graphs. So let's try to see
if we can do that. So let's focus on
this first equation, and I want to graph it. So I have 5x plus
3y is equal to 7. There's a couple of ways
we could graph this. We could put this in
slope-intercept form, or we could just pick
some points here. You just really need two
points to graph a line. So let me just set
some points over here. Let's say x and y. When x is equal to 0, what
does y need to be equal to? So when x is equal
to 0, we have-- let me do it over here-- we
have 5 times 0, plus 3 times y, is equal to 7. That's just 0 over there. So you have 3y is equal to 7. Divide both sides
by 3, you get y is equal to 7/3, which is
the same thing as 2 and 1/3 if we want to write
it as a mixed number. Now let's set y equal to 0. So if we set y as equal to
0, we get 5x, plus 3 times 0, is equal to 7. Or in this part right over
here, it just becomes 0. So we have 5x is equal to 7. Divide both sides
by 5, and we get x equals 7/5, which is the
same thing as 1 and 2/5. So let's graph both
of these points, and then we should be able to
graph this line, or at least a pretty good
approximation of that line. So we have the
point, 0, 2 and 1/3. So that's that point
right over there. So I'll call it 0,
7/3 right over there, and then we have the point,
7/5, 0, or 1 and 2/5, 0. So 1 and 2/5. 2/5 is a little
less than a half. So 1 and 2/5, 0. So our line is going to
look something like this. I just have to connect the dots. It's always hard to
draw the straight line. I'll draw it as a dotted line. So it would look
something like this. Normally, when you have to
solve a system of equations by graphing, they normally
give you a little bit cleaner numbers, but we'll
try our best and see if we can see where these
two lines intersect. So now, let's worry about the
second one right over there. So we have 3x minus
2y is equal to 8. So I'll do the same thing. So 3x minus 2y is equal to 8. We'll just look at the
x- and the y-intercepts. So first, the y-intercept. When x is equal to 0, this whole
thing boils down to 3 times 0 minus 2y is equal to 8. That's just 0. So you have negative
2y is equal to 8. Divide both sides
by negative 2, we get y is equal to negative 4. So the y-intercept
is 0, negative 4. Right over here, and we
mark it 0, negative 4. And then, let's
set y equal to 0. So when y is equal to 0,
this term right over here just becomes 0. So we get 3x minus 2
times 0, so that's just 0. So 3x is equal to 8. Divide both sides by 3,
you get x is equal to 8/3. And 8/3 is the same
thing as 2 and 2/3. So it puts it right about there. That's the point, 0, 8/3. Now let me try my
best to graph it. Connect these two dots. So let me do my very best. I drew a dotted line there. It goes something like that. And just eyeballing it, it looks
like these two lines intersect right over there. I'm hoping that this will
give us a clean answer. And this is the
point 2, negative 1. So that is the
point, 2, negative 1. Right? The x value here is 2.
y value is negative 1. Now, that's what we got
just by eyeballing it, and clearly these are
hand-drawn graphs. Not very precise. Let's verify or let's
see if 2, negative 1 does satisfy both
of these equations, if it's an x and y value
that satisfies both and lies on both the graphs. So if you put 2, negative
1 in this first equation, you get 5 times 2, plus
3 times negative 1, and we're going to see
if that is equal to 7. So this is 10 plus negative 3. So that's the same
thing as 10 minus 3. Does that equal 7? And yeah, it does. 10 minus 3 is equal to 7. So 2, negative 1 is
definitely on that graph or definitely satisfies
that equation. And then, let's do it
with the other one. So if we do 2, negative
1, you have 3 times 2, minus 2 times negative 1. And we're testing to see
if that is equal to 8. So 3 times 2. 3 times 2 is 6. And then 2 times
negative 1 is negative 2. But then, we're subtracting
that, so it's 6 plus 2. 6 plus 2 is equal to 8. And it definitely does equal 8. So we have the coordinate
or we have the point, 2, negative 1 satisfying
both equations. So we've solved the system of
linear equations by graphing.