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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.
(x + 2)
(1 + 2)
