Systems of Equations

We will be looking systems of two equations and two variables, but the equations will not be strictly of lines. Remember that when we find the solutions, they are the (x, y) that work in both equations and geometrically that means they are the points that are on both graphs, meaning the points of intersection of the graphs.

Let’s look at the graphs of these equations: This is a basic parabola moved up 1 This is a line with slope 4 and y int. 1

y



x

2  1

y

 4

x

 1

These graphs intersect in two places so when we algebraically solve this we can expect two answers.

y



x

2  1

y

 4

x

 1

(4, 17) These equations would be easy to solve with substitution. The sub in here to find y the second equation.

x

2  1  4

x

 1

x x

 2

x

  4 4

x

   0 0

(0, 1) This is a quadratic equation so get everything on one side = 0 and factor if possible.

x

 0 ,

x

 4

y

 1

y

 17

x

2 

y

2  10

y



x

 2

A circle centered at (0, 0) with radius square root of 10 A line with slope 1 and y int. 2 Let’s identify and graph each of these equations. What do they look like?

From the second equation we know what y equals so let’s sub it in the first equation.

x

2  

x

 2  2  10

FOIL this

x

2 

x

2  4

x

 4  10

x

2

y

2 

x

 

x y

2   10 2  3 

x x

2   1  2 2  

x x

0 2 2  

x

2  4

x

 4 4

x

2

x

  6 3    0 0  10

x y

   3 ,  1

x y

  1 3

(-3, -1) (1, 3)

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com

and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au