1. 1.4 Solving Linear Systems

2. 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

3. 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.

4. 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.

5. 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.

6. 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 lawn seats.