# Honors Algebra 2: Systems of 3 Linear Equations: Dependent or Inconsistent Systems

• Class: Honors Algebra 2
• Author: Peter Atlas
• Text: Algebra and Trigonometry: Structure and Method, Brown

Calculator: You may use a calculator for help with the arithmetic, but you may not use a systems program or matrices.

Each of the systems below is either inconsistent, or dependent. If it is inconsistent, so state. If it is dependent, characterize the solution set.

1. $$x + 2y + z = 1 \\ -x - y + 2z = 0 \\ y + 3z = 4$$
2. Solution Inconsistent
3. $$x + 2y + z = 1 \\ x - y + z = 1 \\ 2x + y + 2z = 2$$
4. Solution $$\displaystyle \left\{ \left(x, y, z \right) \in \mathbb{R} | \left( 1 - z, 0, z \right) \right\}$$
5. $$x + 2y + z = 1 \\ 3x + 3y + z = 2 \\ 2x + y = 2$$
6. Solution Inconsistent
7. $$x + y + 2z = 1 \\ x - y + z = 1 \\ 2x + 3z = 2$$
8. Solution $$\displaystyle \left\{ \left(x, y, z \right) \in \mathbb{R} | \left( \frac{2 - 3z}{2}, \frac{z}{2}, z \right) \right\}$$
9. $$x + y = 9 \\ y + z = 7 \\ x - z = 2$$
10. Solution Dependent. A characterization of the solution would be $$\displaystyle \{ (x, y, z) \in \mathbb{R} | (2 + z, 7 - z, z) \}$$
11. $$2y + z = 3(-x + 1) \\ x - 3y + z = 4 \\ -2(3x + 2y + z) = 1$$
12. Solution Inconsistent
13. $$3x + 2y + z = 3 \\ x - 3y + z = 4 \\ -6x - 4y - 2z = 1$$
14. Solution Inconsistent
15. $$x + 2y + 4z = 3 \\ 4x - 2y - 6z = 2 \\ \displaystyle x - \frac{y}{2} - \frac{3z}{2} = \frac{1}{2}$$
16. Solution Dependent. A characterization of the solution would be $$\displaystyle \left\{ \left(x, y, z \right) \in \mathbb{R} | \left( \frac{2}{5}z + \frac{5}{2}, \frac{7 - 7z}{5}, z \right) \right\}$$