Quadratic Systems  
 
Where are you guys?


 


I´m at the intersection of 3rd Street and Highway 4.
I´m at one of the intersections of Reno Street and Highway 4.
I´m on Carson Street, close to Highway 4.

In this example, Highway 4 is shaped like a quadratic equation. Each of the streets looks like a linear equation. Solving a quadratic system consisting of a quadratic equation and a linear equation is very similar to finding the location, that is, looking for the intersection point(s). This example covers three unique cases, one location (one solution), two locations (two solutions), and no specific location (no solution).


A quadratic system consists of two equations in two variables to the second degree. There are essentially two algebraic methods for solving them: substitution and elimination. Which one to use depends on the problem at hand. For example, the following system is more easily solved using the substitution method:

x2 + 4xy + y2 = 1
x + y = 1 		(1)
(2)

First, use equation (2) to solve for x:

x = 1 - y (3)

Next, substitute x into equation (1):

(1-y)2 + 4 (1-y)y + y2 = 1

y2 - y = 0

y(y-1) = 0

y1 = 0
y2 = 1

(4)
(5)

Finally, substituting y1 into equation (3), we have:

x1 = 1 - y = 1 - 0 = 1

Substituting y2 into equation (3), we obtain:

x2 = 1 - y2 = 1 - 1 = 0

Therefore, the solution is

x1 = 1
y1 = 0
or        
x2 = 0
y2 = 1

Sometimes, the elimination method works better, as in the following example:

x2 + 3y2 = 7
2x2 - y2 = 7 		(1)
(2)

2(1) - (2):

7y2 = 7

y2 = 1

y1 = 1
y2 = -1

(3)
(4)

Now, substituting y1 = 1 into equation (1), we have:

x2 + 3 12 = 7

x2 = 4

x1 = 2
x2 = -2

(5)
(6)


Now, substituting y2 = -1 into equation (1), we have:

x2 + 3 (-1)2 = 7

x2 = 4

x3 = 2
x4 = -2

(7)
(8)

Therefore, the solution is:

x = x1 = 2
y = y1 = 1
or        
x = x2 = -2
y = y1 = 1
or        
x = x3 = 2
y = y2 = -1
or        
x = x4 = -2
y = y2 = -1

Solved graphically, the above four ordered pairs represent the four intersecting points of equation (1) and (2). Only at these intersecting points, can the ordered pairs satisfy both equations. That is why the intersecting points are the solution of the quadratic system.