Systems of Equations
We have worked with two types of equationsequations 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 underdeterminedthere 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:
(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 intersectionthe points which lie on both linesare 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.