Solving a System of Linear Equations Graphically
Two linear equations of the form ax + by = c, where a, b, and c are real numbers, is called a system of linear equations. When the two lines intersect, the coordinates of the point of intersection are called the solution of the system. The system is then said to be independent and consistent . When the lines are parallel , there is no point of intersection; hence, there is no solution for the system. In this case the system is said to be inconsistent . When the two lines coincide, every point on the line is a solution. The system is said to be dependent (see Fig. 1126 ).
To find a solution to a system of linear equations, plot the graphs for the lines and find the point of intersection.
Examples
Example 1
Find the solution of:
x + y = 8
x  y = 2
Solution 1
Find two points on each line. For x + y = 8, use (7, 1) and (3, 5). For x – y = 2, use (3, 1) and (6, 4). Graph both lines and then find the point of intersection (see Fig. 1127 ).
The point of intersection is (5, 3).
Math Note: To check, substitute the values of x and y for the solution in both equations and see if they are closed true equations.
Example 2
Find the solution of:
3x  y = 8
x + 2y = 5
Solution 2
Find two points on each line. For 3x – y = 8, use (2, –2) and (4, 4). For x + 2y = 5, use (7, –1) and (1, 2). Graph both lines and then find the point of intersection (see Fig. 1128 ).
The point of intersection is (3, 1).
Solving a System of Linear Equations Graphically Practice Problems
Practice
Find the solution for each of the following system of equations:

x  y = 4
x + y = 10

2x + 3y = 12
x + 4y = 11

4x + y = 10
2x + y = 6

x – 5y = 6
5x – y = 6

x + 3y = –7
6x – y = 15
Answers
 (7, 3)
 (3, 2)
 (2, 2)
 (1, 1)
 (2, 3)
Practice problems for these concepts can be found at: Graphing Practice Test.
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