Main content

## 8th grade (Eureka Math/EngageNY)

### Unit 4: Lesson 4

Topic D: Systems of linear equations and their solutions- Systems of equations: trolls, tolls (1 of 2)
- Systems of equations: trolls, tolls (2 of 2)
- Testing a solution to a system of equations
- Solutions of systems of equations
- Systems of equations with graphing
- Systems of equations with graphing
- Systems of equations with graphing: 5x+3y=7 & 3x-2y=8
- Systems of equations with graphing: y=7/5x-5 & y=3/5x-1
- Systems of equations with graphing: chores
- Systems of equations with graphing
- Systems of equations with elimination: 3t+4g=6 & -6t+g=6
- Systems of equations with elimination
- Systems of equations with elimination: x+2y=6 & 4x-2y=14
- Systems of equations with elimination: -3y+4x=11 & y+2x=13
- Systems of equations with elimination: 2x-y=14 & -6x+3y=-42
- Systems of equations with elimination: 4x-2y=5 & 2x-y=2.5
- Systems of equations with elimination: x-4y=-18 & -x+3y=11
- Systems of equations with elimination
- Systems of equations with elimination: 6x-6y=-24 & -5x-5y=-60
- Systems of equations with elimination challenge
- Systems of equations with substitution: 2y=x+7 & x=y-4
- Systems of equations with substitution
- Systems of equations with substitution: y=4x-17.5 & y+2x=6.5
- Systems of equations with substitution: -3x-4y=-2 & y=2x-5
- Systems of equations with substitution: 9x+3y=15 & y-x=5
- Systems of equations with substitution
- Systems of equations with substitution: y=-5x+8 & 10x+2y=-2
- Systems of equations with substitution: y=-1/4x+100 & y=-1/4x+120
- Systems of equations with elimination: apples and oranges
- Systems of equations with elimination: TV & DVD
- Systems of equations with elimination: King's cupcakes
- Systems of equations with elimination: Sum/difference of numbers
- Systems of equations with elimination: potato chips
- Systems of equations with elimination: coffee and croissants
- Systems of equations with substitution: coins
- Systems of equations with substitution: potato chips
- Systems of equations with substitution: shelves
- Systems of equations word problems
- Age word problem: Imran
- Age word problem: Ben & William
- Age word problem: Arman & Diya
- Age word problems
- Solutions to systems of equations: consistent vs. inconsistent
- Systems of equations number of solutions: fruit prices (1 of 2)
- Systems of equations number of solutions: fruit prices (2 of 2)
- Systems of equations number of solutions: y=3x+1 & 2y+4=6x
- Solutions to systems of equations: dependent vs. independent
- Number of solutions to a system of equations graphically
- Number of solutions to a system of equations graphically
- Forming systems of equations with different numbers of solutions
- Number of solutions to a system of equations algebraically
- Comparing Celsius and Fahrenheit temperature scales
- Converting Fahrenheit to Celsius

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Forming systems of equations with different numbers of solutions

Sal write an equation that along with the equation 4x + 5y = 2 forms a system of equations with infinitely many solutions. Created by Sal Khan.

## Want to join the conversation?

- when 0;10 says 'infinitely many solutions'' does it mean the solutions go on forever? Because in doing the problems their is only one way to do them.(9 votes)
- The solution to a system of equations is the point or points at which their graphs intersect. Thus, when you have two ways of expressing the same equation, you get infinitely many intersections (because they are identical). So, every point that lies on the graph of one equation (and there are infinitely many such points) will also lie on the graph of the other.(16 votes)

- Can you have systems of equations with something other than lines?(6 votes)
- Yes. These are called non-linear systems, and when solving them, you are finding the points where other types of graphs intersect each other. These systems can have more than one solution (yet without having an infinite number of solutions as is the case with lines).

For example, imagine the graph of two parabolas -- one pointing up and the other down. You can position these in a way where there are two points of intersection. To find those, you would solve a non-linear system.(9 votes)

- I am having trouble determining how to complete some of the problems in "understanding solution methods to systems of equations" the video linked doesnt really cover this module clearly. In some of the questions it asks you to "add" the equations together, but in the hints it does the problem sometimes by "subtracting" when it tells you to "add", then in other problems it tells you to "add" them together and it does the problem by "adding" them - I dont understand this discrepancy. Thx in advance for any help!(2 votes)
- When using elimination to solve a system of equations, adding OR subtracting the equations together are valid processes. When do you add and when do you subtract?

If the coefficients of the variable you are eliminating are exactly the same, then subtract.

Example...

3x + 5y = 12

3x + 4y = 6

Subtract the equations to get y = 18

If the If the coefficients of the variable you are eliminating are exactly the same but with different signs, then add.

Example

3x + 7y = 15

-3x - 6y = 6

Add the equations together to get y = 21(3 votes)

- HELP! I am stuck on a problem called:

Graphically understanding solution methods to systems of equations.

This is the last question I have to finish before the end of the summer. I have watched all the videos and understand everything except for adding and subtracting the equations. I have tried for five or six days and I can't get it right, I look through the hints and study it all but I cant get it right! No one I have asked in my house can help and I am frustrated. Please help.

-Chance.(2 votes)- Here is a few links...It is a bit difficult to explain so if you look up "Systems of Equations Elimination" om Google, you should see a few websites that explain it fairly thouroughly...

http://www.purplemath.com/modules/systlin5.htm

http://www.mathplanet.com/education/algebra-1/systems-of-linear-equations-and-inequalities/the-elimination-method-for-solving-linear-systems

http://coolmath.com/algebra/12-2x2-systems-of-equations/03-solving-by-elimination-addition-01.htm

I think these websites explain pretty well. Sorry, my advice isn't of too much help. Good luck, though!(3 votes)

- I don't get anything on "graphically understanding solution methods to systems of equation". Can you explain it.(2 votes)
- I don't get any right either! The questions don't relate, and the hints don't make sense!(1 vote)

- I am too confused in this topic in the last question of the video(1 vote)
- At1:28whats the porpose of multipying by 2?(1 vote)
- is a infinite solution consistent/dependent?(1 vote)
- Here is another problem: no where is putting one side of an equation to the other then graphing them (and having a solution between all three equations) taught here. It is also deceptive, because they then tell you that this doesn't work.(1 vote)
- A MISTAKE: Sal wrote the second equation in slope intercept form as

y=6x/7+a/7, but it really is y=6x/7-a/7.

If I am wrong please tell me.(1 vote)

## Video transcript

Fill in the blanks to form
a system of linear equations in the variables x and y with
infinitely many solutions. So you're going to have
infinitely many solutions if essentially both
of these equations are describing the same line. If they're both essentially
the same equation, they are the same constraint. And we can graphically
imagine that. Let's say that that's our
y-axis, and this is our x-axis. And we could even try to
graph this right over here. If we were to put this
into slope-intercept form, you would have 5y is
equal to 2 minus 4x. And then if we divide both sides
by 5, you get y is equal to-- and I'll swap these--
negative 4/5 x plus 2/5. So this line up here is going
to look something like this. It's y-intercept is at 2/5. And it's going to have
a negative 4/5 slope. So it's going to look
something like this. So that's what the
line looks like. So we're going to get an
infinite number of solutions for the system. If the second line,
if when we graph it, is essentially the
exact same line, it overlaps at
every x and y that satisfy the first equation. Now, the easiest way to think
about what these blanks should be is, well, how do
I fill in the blanks here so it is really just a
direct algebraic manipulation of this first equation? And we have a clue here. On the right-hand side of the
first equation, we have a 2. On the right-hand side of the
second equation, we have a 4. So if we want a direct
algebraic manipulation, clearly on the right-hand side,
you have multiplied by 2. And if you want this to
be the same equation, you have to do
that to both sides. So let's multiply
both sides by 2. So 4x times 2 is 8x. 5y times 2 is 10y. So this equation
and this equation are the exact same constraint. They represent the
exact same line. They have an infinite
number of solutions. Let's do another example. Which of the following
choices of a will result in a system of linear
equations with no solutions? So you're going to have
no solutions as if you take the same
combination of x and y's, but you get to a
different number. Or another way of
thinking about it is if you plotted
this equation and it had the exact same slope, but
it had a different y-intercept. So let's think
about it both ways. So first, let's try to
algebraically manipulate the second equation so that
the left-hand side looks exactly the same as the
left-hand side up here. And then we just
have to make sure that the right-hand
side is different. And then you will
have no solutions. So let's think about
that a little bit. So it looks pretty close. If we multiply the left-hand
side by a negative 1, it's going to look just like
the left-hand side over here. So let's do that. Let's multiply the left-hand
and the right-hand side by negative 1. So it's essentially
the same equation. So negative 6x times
negative 1 is positive 6x. 7y times negative 1 is minus 7y. And a times negative
1 is negative a. So notice, now on the
left-hand side we have the same combinations of x's and y's. We have 6 x's minus 7 y's. If on the right-hand side these
two things were to be equal, then we'd have the
scenario that we just saw. We would have the
exact same line. If negative a was equal to 4,
we would have the same line. So let me write that down. If negative a is
equal to 4, then we are dealing with the same
line/equation/constraint. And like the
previous scenario, we would have an infinite
number of solutions. On the other hand, if
negative a is not equal to 4, then there's no way
that there's anything that will satisfy both of them. Here you're saying
that whatever your x is and whatever your y is, take
6 of the x's and subtract 7 of the y's, and you get 4. If here you take 6 of
the x's and subtract 7 of the y's and you get
a different number, then there's never
going to be an x and a y that satisfy
both of them. So if negative 8
is not equal to 4, then you're going to
have no solutions. Or another way of saying this is
if a does not equal negative 4, obviously, if a is
equal to negative 4, then negative a is
going to be equal to 4. So as long as a is not
equal to negative 4, you're going to
have no solutions. So a can be any number
except for negative 4. Now, another way
to think about this is to put them both in
slope-intercept form. And you'll see that they'll
have the same slope. And then you would
just want to have them have different
y-intercepts. And they're going to be parallel
lines that don't overlap. So this top one right over
here, if we subtract 6x from both sides,
you get negative 7y is equal to negative 6x plus 4. Divide both sides by negative 7. You get y is equal
to 6/7 x minus 4/7. So that's this first one
right up here written in slope-intercept form. Now, the second
one, I'll just start with exactly what they gave us. Let me add 6x to both sides. So this is the second one. So if I add 6x to both sides,
I get 7y is equal to 6x plus a. And then if I divide
both sides by 7, I get y is equal to
6/7 x plus a over 7. So notice in slope-intercept
form, our first equation looks like that. And in terms of a, our second
equation looks like that. So they definitely already
have the same slope. They have the exact same slope. If a is equal to
negative 4, then these are going to be the
exact same equations. They're going to have the
exact same y-intercept. You're going to have an
infinite number of solutions. On the other hand, if a is
anything other than negative 4, you're going to have a
different y-intercept. And these two things are
just going to be parallel.