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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:
y=12x+3
y=x+1
First, let's graph the first equation y=12x+3. 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.
A graph of linear function on the coordinate plane. The horizontal x axis runs from negative 10 to 10 in intervals of 1. The vertical y axis runs from negative 10 to 10 in intervals of 1. The function increases as it moves left to right. The function passes through the points (negative 6, 0), (0, 3), and (8, 7). Closed points are plotted at (0, 3) and (2, 4).
Next, let's graph the second equation y=x+1 as well.
A graph of 2 linear functions on the coordinate plane. The horizontal x axis runs from negative 10 to 10 in intervals of 1. The vertical y axis runs from negative 10 to 10 in intervals of 1. The function represented by the orange line increases as it moves left to right. The function passes through the points (negative 6, 0), (0, 3), and (8, 7). Closed points are plotted at (0, 3) and (2, 4). The function represented by the green line increases as it moves left to right. The function passes through the points (negative 1, 0), (0, 1), and (7, 8). The two lines intersect at the point (4, 5).
There is exactly one point where the graphs intersect. This is the solution to the system of equations.
A graph of 2 linear functions on the coordinate plane. The horizontal x axis runs from negative 10 to 10 in intervals of 1. The vertical y axis runs from negative 10 to 10 in intervals of 1. The function represented by the orange line increases as it moves left to right. The function passes through the points (negative 6, 0), (0, 3), and (8, 7). Closed points are plotted at (0, 3) and (2, 4). The function represented by the green line increases as it moves left to right. The function passes through the points (negative 1, 0), (0, 1), and (7, 8). The two lines intersect at the point (4, 5). There is a closed point labeled (4, 5) at the point where the lines intersect.
This makes sense because every point on the gold line is a solution to the equation y=12x+3, and every point on the green line is a solution to 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) is the solution to the system. Let's verify this by plugging x=4 and y=5 into each equation.
The first equation:
y=12x+35=?12(4)+3Plug in x = 4 and y = 55=5Yes!
The second equation:
y=x+15=?4+1Plug in x = 4 and y = 55=5Yes!
Nice! (4,5) is indeed a solution.

Let's practice!

Problem 1

The following system of equations are graphed below.
y=3x7
y=x+9
A graph of 2 linear functions on the coordinate plane. The horizontal x axis runs from negative 10 to 10 in intervals of 1. The vertical y axis runs from negative 10 to 10 in intervals of 1. The function represented by the orange line decreases as it moves left to right. The function passes through the points (negative 5, 8), (negative 3, 2), and (0, negative 7). The function represented by the green line increases as it moves left to right. The function passes through the points (negative 9, 0), (negative 6, 3), and (0, 9). The two lines intersect at the point (negative 4, 5).
Find the solution to the system of equations.
x=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
y=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Problem 2

Here is a system of equations:
y=5x+2
y=x+8
Graph both equations.
Find the solution to the system of equations.
x=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
y=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Problem 3

Here is a system of equations:
8x4y=16
8x+4y=16
Graph both equations.
Find the solution to the system of equations.
x=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
y=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Challenge problems

1) How many solutions does the system of equations graphed below have?
Choose 1 answer:
A graph of 2 linear functions on the coordinate plane. The horizontal x axis runs from negative 10 to 10 in intervals of 2. The vertical y axis runs from negative 10 to 10 in intervals of 2. The function represented by the orange line decreases as it moves left to right. The function passes through the points (negative 6, 8), (negative 1, 0), and (0, negative 1). The function represented by the green line increases as it moves left to right. The function passes through the points (negative 3, 0), (3, 0), and (3, 6). The lines intersect at an x-value between negative 2 and 0 and a y-value between 0 and 2.

2) How many solutions does the system of equations graphed below have?
(The two lines are parallel, so they never intersect)
Choose 1 answer:
A graph of 2 linear functions on the coordinate plane. The horizontal x axis runs from negative 10 to 10 in intervals of 2. The vertical y axis runs from negative 10 to 10 in intervals of 2. The function represented by the orange line increases as it moves left to right. The function passes through the points (negative 6, negative 7), (negative 1, 3), and (0, 5). The function represented by the green line increases as it moves left to right. The function passes through the points (0, negative 1), (2, 3), and (4, 7). The lines are parallel.

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:
A graph of 2 linear functions on the coordinate plane. The horizontal x axis runs from negative 10 to 10 in intervals of 2. The vertical y axis runs from negative 10 to 10 in intervals of 2. The function represented by the orange line increases as it moves left to right. The function represented by the green line increases as it moves left to right. The function passes through the points (negative 6, 0), (0, 2), and (6, 4). The lines are parallel.

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:

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