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Section 2.1: Solving Systems of Equations by
Graphing
Objective: Solve systems of equations by graphing and identifying the point of
intersection.
We have solved equations like (^3) x 4 11 by adding 4 to both sides and then dividing by 3
(solution is x 5 ). We also have methods to solve equations with more than one variable in
them. It turns out that to solve for more than one variable we will need the same number of
equations as variables. For example, to solve for two variables such as x and y we will need
two equations. When we have several equations we are using to solve, we call the equations
a system of equations. When solving a system of equations we are looking for a solution
that works for all of these equations. In this discussion, we will limit ourselves to solving two
equations with two unknowns. This solution is usually given as an ordered pair ( , x y^ ). The
following example illustrates a solution working in both equations.
Example 1.
Show (2,1)is the solution to the system
x y x y
(2,1) Identify x and y from the ordered pair
x 2 , y 1 Plug these values into each equation
First equation Evaluate True
Second equation, evaluate True
As we found a true statement for both equations we know (2,1)is the solution to the system.
It is in fact the only combination of numbers that works in both equations. In this lesson we
will be working to find this point given the equations. It seems to follow that if we use points
to describe the solution, we can use graphs to find the solutions.
If the graph of a line is a picture of all the solutions, we can graph two lines on the same
coordinate plane to see the solutions of both equations. We are interested in the point that is
a solution for both lines, this would be where the lines intersect! If we can find the intersection
of the lines we have found the solution that works in both equations.
Example 2.
Solve the system of equations by graphing:
1 2 3 4
y x
y x
y x To graph we identify slopes and y - intercepts
3 2 4
y x
First:
m , b 3
Second:
m , b 2
Now we can graph both lines on the same plane.
To graph each equation, we start at the y - intercept and use the slope rise run^ to get the next point and connect the dots.
Remember a negative slope is downhill!
Find the intersection point,(4,1)
(4,1) Our Solution
Often the equations will not be in slope-intercept form. We can solve both equations for y
first to put the equation in slope-intercept form.
Example 3.
Solve the system of equations by graphing:
x y x y
x y x y
Solve each equation for y
6 3 9 6 6 3 6 9
x y x x y x
x y x x y x
Subtract x terms Put x terms first
(4,1)
You can graph lines using intercepts as well.
Example 5.
Solve the system of equations by graphing:
x y x y
x y x x x
Let us graph using intercepts.
x - intercept (let y 0 )
(6, 0) Our x - intercept
y y y
(0, 2) Our y - intercept
x y x x x
Let us graph using intercepts again.
x - intercept (let y 0 )
(6, 0) Our x - intercept
y y y
(0, 2 ) Our y - intercept
Both equations are the same line! As one line is directly on top of the other line, we can say that the lines “intersect” at all the points! Here we say we have infinite solutions.
World View Note: The Babylonians were the first to work with systems of equations with
two variables. However, their work with systems was quickly passed by the Greeks who
would solve systems of equations with three or four variables and, around 300 AD, developed
methods for solving systems with any number of unknowns!
2.1 Practice
Solve each system of equations by graphing.
y x y x
5 4 1 4 2
2
y x
y x
y y x
3
y x
y x
3 4 3 4
y x
y x
y x y x
1 3 5 3
y x
y x
2
y x
y x
5 3 2 3
y x
y x
1 2 1 2
y x
y x
x y x y
y x y x
y x x y
x y x y
x y y x
2.1 Answers
1) ( 1,^2 )
- No Solution
- No Solution
14)( 1,^3 )
15)(3,^ 4)
- No Solution
17)(2,^ 2)
- Infinite Solutions
- No Solution
- Infinite Solutions
