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Systems of Equations

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Systems of Equations

We have worked with two types of equations--equations with one variable and equations with two variables. In general, we could find a limited number of solutions to a single equation with one variable, while we could find an infinite number of solutions to a single equation with two variables. This is because a single equation with two variables is underdetermined--there are more variables than equations. But what if we added another equation?

A system of equations is a set of two or more equations with the same variables. A solution to a system of equations is a set of values for the variable that satisfy all the equations simultaneously. In order to solve a system of equations, one must find all the sets of values of the variables that constitutes solutions of the system.

Example: Which of the ordered pairs in the set {(5, 4),(3, 8),(6, 4),(4, 6),(7, 2)} is a solution of the following system of equations:

y + 2x = 14  
xy = 24  

(5, 4) is a solution of the first equation, but not the second.
(3, 8) is a solution of both equations.
(6, 4) is a solution of the second equation, but not the first.
(4, 6) is a solution of both equations.
(7, 2) is not a solution of either equation.
Thus, the solution set of the system is {(3, 8),(4, 6)} .

Solving Systems of Linear Equations by Graphing

When we graph a linear equation in two variables as a line in the plane, all the points on this line correspond to ordered pairs that satisfy the equation. Thus, when we graph two equations, all the points of intersection--the points which lie on both lines--are the points which satisfy both equations.

To solve a system of equations by graphing, graph all the equations in the system. The point(s) at which all the lines intersect are the solutions to the system.

Example: Solve the following system by graphing:

y - 3 = - (x + 2)  
y = 3x - 2  

Graph of System
Since the two lines intersect at the point (1, 1) , this point is a solution to the system. Thus, the solution set to the system of equations is {(1, 1)} .

To check, plug (1, 1) in to both equations:
1 - 3 = - (1 + 2) ? Yes.
1 = 3(1) - 2 ? Yes.

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