CCSS Math: 8.EE.C.8, 8.EE.C.8a
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:
y=12x+3\goldD{y=\dfrac{1}{2}x+3}
y=x+1\greenE{y=x+1}
First, let's graph the first equation y=12x+3\goldD{y=\dfrac{1}{2}x+3}. Notice that the equation is already in yy-intercept form so we can graph it by starting at the yy-intercept of 33, and then going up 11 and to the right 22 from there.
Next, let's graph the second equation y=x+1\greenE{y=x+1} 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 y=12x+3\goldD{y=\dfrac{1}{2}x+3}, and every point on the green line is a solution to y=x+1\greenE{y=x+1}. 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 (4,5)(4,5) is the solution to the system. Let's verify this by plugging x=4x =4 and y=5y = 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! (4,5)(4, 5) is indeed a solution.

Let's practice!

Problem 1

The following system of equations are graphed below.
y=3x7y=-3x-7
y=x+9y=x+9
Find the solution to the system of equations.
x=x =
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}
y=y =
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

When we graph the two lines, we can see that they intersect at (4,5)(-4,5). Therefore, this is the solution of the system:
  • x=4x=-4
  • y=5y=5

Problem 2

Here is a system of equations:
y=5x+2y=5x+2
y=x+8y=-x+8
Graph both equations.
Find the solution to the system of equations.
x=x=
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}
y=y=
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

When we graph the two lines, we can see that they intersect at (1,7)(1,7). Therefore, this is the solution of the system:
  • x=1x=1
  • y=7y=7

Problem 3

Here is a system of equations:
8x4y=168x-4y=16
8x+4y=168x+4y=16
Graph both equations.
Find the solution to the system of equations.
x=x=
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}
y=y=
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

When we graph the two lines, we can see that they intersect at (2,0)(2,0). Therefore, this is the solution of the system:
  • x=2x=2
  • y=0y=0

Challenge problems

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

The system of equations has 11 point of intersection so there is 11 solution.
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:

The system of equations has 00 points of intersection so there are 00 solutions.
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:

The system of equations has infinite points of intersection so there are infinite solutions.
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:

As we saw in the last three problems, two lines can either intersect at 00, 11, or infinite points. It is impossible for two lines to intersect at 22 points.
No, a system of linear equations cannot have exactly 22 solutions.