1.4 Solving Linear Systems

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Presentation transcript:

1.4 Solving Linear Systems

Visualizing solutions of systems Linear equation of three variables (x,y,z) follows the form: ax+by+cz=d A solution of a system of these equations is an ordered triple (x, y, z). The graph of a linear equation in three variables is a plane in three-dimensional space. Exactly one solution Infinitely many solutions No solution The planes intersect in a single point The planes intersect in a line No points in common. The planes do not intersect

Example 1 Solve the system of equations: Rewrite as a system of two equations with variables. 2. Solve the new linear system for both variables. 3. Substitute into any original equation and solve for y.

Example 2 Solve the system of equations 1. Rewrite the system as a linear system in two variables. Because this is a false equation, the original system has no solution.

Example 3 Solve the system of equations 1. Rewrite the system as a linear system in two variables. 2. Solve the new linear system for both of its variables. The system has infinitely many solutions.

Example 4 An amphitheater charges $75 for each seat in Section A, $55 for each seat in Section B, and $30 for each lawn seat. There are three times as many seats in Section B as in Section A. The revenue from selling all 23,000 seats is $870,000. How many seats are in each section of the amphitheater? Variables: x=number of seats in section A, y= number of seats in B, z= number of lawn seats 1. Write a system of equations. 2. Rewrite as a linear system in two variables by substituting 3x for y. There are 1500 seats in Section A, 4500 seats in Section B, and 17000 lawn seats.