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8th grade (Eureka Math/EngageNY)

Unit 4: Lesson 4

Topic D: Systems of linear equations and their solutions

Systems of equations with graphing

Walk through examples of solving systems of equations by finding the point of intersection.
We can find the solution to a system of equations by graphing the equations. Let's do this with the following systems of equations:
start color #e07d10, y, equals, start fraction, 1, divided by, 2, end fraction, x, plus, 3, end color #e07d10
start color #0d923f, y, equals, x, plus, 1, end color #0d923f
First, let's graph the first equation start color #e07d10, y, equals, start fraction, 1, divided by, 2, end fraction, x, plus, 3, end color #e07d10. Notice that the equation is already in y-intercept form so we can graph it by starting at the y-intercept of 3, and then going up 1 and to the right 2 from there.
Next, let's graph the second equation start color #0d923f, y, equals, x, plus, 1, end color #0d923f as well.
There is exactly one point where the graphs intersect. This is the solution to the system of equations.
This makes sense because every point on the gold line is a solution to the equation start color #e07d10, y, equals, start fraction, 1, divided by, 2, end fraction, x, plus, 3, end color #e07d10, and every point on the green line is a solution to start color #0d923f, y, equals, x, plus, 1, end color #0d923f. So, the only point that's a solution to both equations is the point of intersection

Checking the solution

So, from graphing the two equations, we found that the ordered pair left parenthesis, 4, comma, 5, right parenthesis is the solution to the system. Let's verify this by plugging x, equals, 4 and y, equals, 5 into each equation.
The first equation:
y=12x+35=?12(4)+3Plug in x = 4 and y = 55=5Yes!\begin{aligned} \goldD{y} &\greenE= \goldD{\dfrac12x + 3} \\\\ 5&\stackrel?= \dfrac12(4) + 3 &\gray{\text{Plug in x = 4 and y = 5}}\\\\ 5 &= 5 &\gray{\text{Yes!}}\end{aligned}
The second equation:
y=x+15=?4+1Plug in x = 4 and y = 55=5Yes!\begin{aligned} \greenE{y} &\greenE= \greenE{x+1} \\\\ 5&\stackrel?= 4 + 1 &\gray{\text{Plug in x = 4 and y = 5}}\\\\ 5 &= 5 &\gray{\text{Yes!}}\end{aligned}
Nice! left parenthesis, 4, comma, 5, right parenthesis is indeed a solution.

Let's practice!

Problem 1

The following system of equations are graphed below.
y, equals, minus, 3, x, minus, 7
y, equals, x, plus, 9
Find the solution to the system of equations.
x, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
y, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Problem 2

Here is a system of equations:
y, equals, 5, x, plus, 2
y, equals, minus, x, plus, 8
Graph both equations.
Find the solution to the system of equations.
x, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
y, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Problem 3

Here is a system of equations:
8, x, minus, 4, y, equals, 16
8, x, plus, 4, y, equals, 16
Graph both equations.
Find the solution to the system of equations.
x, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
y, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Challenge problems

1) How many solutions does the system of equations graphed below have?
Choose 1 answer:
Choose 1 answer:

2) How many solutions does the system of equations graphed below have?
(The two lines are parallel, so they never intersect)
Choose 1 answer:
Choose 1 answer:

3) How many solutions does the system of equations graphed below have?
(The two lines are exactly the same. They are directly on top of each other, so there are an infinite number of points of intersection.)
Choose 1 answer:
Choose 1 answer:

4) Is it possible for a system of linear equations to have exactly two solutions?
Hint: Think about the graphs in the problems above.
Choose 1 answer:
Choose 1 answer:

Want to join the conversation?

  • blobby green style avatar for user tkaufmann
    Is there some potion or something I can drink to become better and have MEMORY RETENTION when it comes to math? I am in college and this is all new to me. I moved around alot when younger, so I never really got the concept of more advanced mathematics. I just stare at problems for long times and have no idea where to start half the time, but once I get rolling I don't stop. Any advice?
    (23 votes)
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  • leafers ultimate style avatar for user Ivan
    How could you convert a normal system of equations into slope intercept form?
    (7 votes)
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    • marcimus purple style avatar for user Evan Evan
      You've waited 5 years for the answer, so here it is. You can solve one equation for either variable, then plug that into the other equation and solve that one completely, plug that back into the first equation, and now you know all the variables and you can do whatever you want with them, like put them into slope intercept form.
      (8 votes)
  • boggle blue style avatar for user x.asper
    Please help me! I have no idea how to even find out which coordinates I am supposed to put these lines on. They don't explain this concept well.
    (13 votes)
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    • blobby green style avatar for user JANESHB
      When given a Slope-Intercept form equation what you can do to graph is graph the y-intercept "b" which is the number without a variable on the graph first, this means you have to put that point on the vertical line where the "b" is shown. Then you just have to move one on the x-axis (the horizontal) and the amount next to the x up or down (the number next to the x is the "m" better known as the slope). I hope this helps.
      (2 votes)
  • blobby green style avatar for user McCoy, Hayden
    i litterally have no clue what im doing rn!
    (8 votes)
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  • stelly yellow style avatar for user RareWare
    Is there anyone who can explain this to me?

    I'm in 8th grade. My teacher uses a service called "my.hrw" and every time you get it wrong, it never explains why. I know adults say "ask the teacher" but he always explains like I know what he's saying. There are lots of things about this and all the adults I ask about math in general get angrier every time I ask a question like "ok, but why are we doing this" or "why do we use THIS number instead of THAT number." I've asked the teacher to explain it to me multiple times, I've watched almost every YouTube and Khan Academy video at this point, and looked on every website imaginable. It seems like everyone else gets this concept except I. I've got to learn this to get to the next thing that I'm supposed to be learning right now but I can't grasp that concept because it requires you know how to do this! Can ANYONE explain this in simple, easy to understand terms, and break it down step by step, process by process?
    (10 votes)
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    • purple pi pink style avatar for user funDAmental
      Well, if you're talking about the x and y-intercepts, I guess this will help (btw im a 7th grader so plz correct me if im using anything wrong hehe)-

      In every equation, e.g. y=2x+1 , they will give you intercepts like (1,1) etc. What you have to do here is-
      1. Plug in the x and y's into the equation (like since it is 2x then you will be doing 2*1)
      2. Do the operations (like here, you will do 2-1)
      3. See if both the sides have the same number. If not, the given intercepts are not the correct answer. If yes, then it will be the answer.

      It will go like this-
      y=2x-1
      1 (because y is 1)=(2*1)-1
      1=2-1
      1=1

      If you get this, you're good to go!! Hope this helps:)
      (1 vote)
  • piceratops seed style avatar for user Nik Miller
    it was really hard to understand
    (10 votes)
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  • blobby green style avatar for user cl04291
    can a system of equations be be in multiple ways
    (8 votes)
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  • leafers tree style avatar for user Ringo, Emmett
    why y=Mx+b i doesn't make sense
    (4 votes)
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  • blobby green style avatar for user coltsfugate
    If you are a science person and you try look on the graph how much is it on there
    (4 votes)
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  • blobby green style avatar for user M.A.R.I.A
    That was confusing but i started to get in the end.
    (4 votes)
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