Systems of Linear Equations 7.1 Systems of Linear Equations

System of Linear Equations A system of linear equations (or simultaneous linear equations) is two or more linear equations. A solution to a system of equations is the ordered pair or pairs that satisfy all equations in the system. A system of linear equations may have exactly one solution, no solution, or infinitely many solutions.

Example: Is the Ordered Pair a Solution? Determine which of the ordered pairs is a solution to the following system of linear equations. 3x – y = 2 4x + y = 5 a. (2, 4) b. (1, 1)

Example: Is the Ordered Pair a Solution? continued Check (2, 4): 3x – y = 2 4x + y = 5 3(2) – 4 = 2 4(2) + 4 = 5 2 = 2 True 12 = 5 False Since (2, 4) does not satisfy both equations, it is not a solution to the system.

Example: Is the Ordered Pair a Solution? continued Check (1, 1): 3x – y = 2 4x + y = 5 3(1) – 1 = 2 4(1) + 1 = 5 2 = 2 True 5 = 5 True Since (1, 1) satisfies both equations, it is a solution to the system.

Practice Problem: Determine which ordered pair is the solution. 3x + y = 5 x – y = 7 a. ( 2, -1) b. (4, -3) c. (3, -4)

Procedure for Solving a System of Equations by Graphing Determine three ordered pairs that satisfy each equation. Plot the ordered pairs and sketch the graphs of both equations on the same axes. The coordinates of the point or points of intersection of the graphs are the solution or solutions to the system of equations.

Graphing Linear Equations When graphing linear equations, three outcomes are possible: 1. The two lines may intersect at one point, producing a system with one solution. A system that has one solution is called a consistent system of equations. 2. The two lines may be parallel, producing a system with no solutions. A system with no solutions is called an inconsistent system.

Graphing Linear Equations continued 3. The two equations may represent the same line, producing a system with infinite number of solutions. A system with an infinite number of solutions is called a dependent system.

Example Find the solution to the following system of equations graphically. x – 3y = 9 2x – 6y = 3 Solution: Since the two lines are parallel, they do not intersect, therefore, the system has no solution.

Practice Problem: Solve by graphing

Homework P. 391 # 12 – 46 even