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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:
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:

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