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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 |
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 |