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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:
\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:
\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
y, equals

### 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
y, equals

### 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
y, equals

## Challenge problems

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

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

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.)

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

## Want to join the conversation?

• 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?
• How long have you been on here? Just curious...
• How could you convert a normal system of equations into slope intercept form?
• 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.
• 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.
• 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.
(1 vote)
• it was really hard to understand
• *If I am being frank, this is very hard, I took around 2-3 hours of work to understand. Just keep on going.*
• why y=Mx+b i doesn't make sense
• m is the slope and b is the y-intercept and then x and y are just the x and y coordinates