Title: Solve Linear Systems

Q: Created by: Dr. Pang Chung Yin Amy, Dr. Chiang Shun Ling, and Naw Sai.
Department of Mathematics,
Hong Kong Baptist University (HKBU) © 2025.
A: These decks may be freely shared and distributed among students and educators. We hope they help in your linear algebra journey!

Q: Solve the following linear system:
$2x_{1} + x_{2} = 6$
$-x_{1} - x_{2} = -3$
A: $x_{1} = 3$
$x_{2} = 0$

Q: Solve the following linear system:
$-x_{1} - 2x_{2} = -3$
$-x_{1} - 6x_{2} = -11$
A: $x_{1} = -1$
$x_{2} = 2$

Q: Solve the following linear system:
$4x_{1} + 5x_{2} = 8$
$5x_{1} + 7x_{2} = 10$
A: $x_{1} = 2$
$x_{2} = 0$

Q: Solve the following linear system:
$-x_{1} + 2x_{2} = -3$
$-3x_{1} + 2x_{2} = -5$
A: $x_{1} = 1$
$x_{2} = -1$

Q: Solve the following linear system:
$12x_{1} + 13x_{2} = 2$
$-6x_{1} - 7x_{2} = -2$
A: $x_{1} = -2$
$x_{2} = 2$

Q: Solve the following linear system:
$2x_{1} + x_{2} = 5$
$5x_{1} + 3x_{2} = 14$
A: $x_{1} = 1$
$x_{2} = 3$

Q: Solve the following linear system:
$3x_{1} + 2x_{2} = -3$
$x_{1} + 2x_{2} = 3$
$3x_{1} + 2x_{2} = -3$
A: $x_{1} = -3$
$x_{2} = 3$

Q: Solve the following linear system:
$3x_{1} - x_{2} = 0$
$x_{1} + x_{2} = 0$
$-x_{1} + x_{2} = 0$
A: $x_{1} = 0$
$x_{2} = 0$

Q: Solve the following linear system:
$x_{1} + x_{2} = -1$
$-2x_{1} + x_{2} = 2$
$2x_{1} - x_{2} = -2$
A: $x_{1} = -1$
$x_{2} = 0$

Q: Solve the following linear system:
$-x_{1} + 3x_{2} = -9$
$x_{1} - 3x_{2} = 9$
$-x_{1} + 4x_{2} = -11$
A: $x_{1} = 3$
$x_{2} = -2$

Q: Solve the following linear system:
$2x_{1} - x_{3} = 2$
$x_{1} + x_{2} - x_{3} = 2$
$x_{1} - x_{3} = 2$
A: $x_{1} = 0$
$x_{2} = 0$
$x_{3} = -2$

Q: Solve the following linear system:
$-5x_{1} + 6x_{2} = -4$
$-x_{1} + 3x_{2} + x_{3} = 0$
$2x_{1} + x_{3} = 3$
A: $x_{1} = 2$
$x_{2} = 1$
$x_{3} = -1$

Q: Solve the following linear system:
$x_{1} + x_{3} = -4$
$3x_{1} + 2x_{2} + 3x_{3} = -8$
$2x_{1} + 3x_{3} = -10$
A: $x_{1} = -2$
$x_{2} = 2$
$x_{3} = -2$

Q: Solve the following linear system:
$x_{1} - 2x_{3} = 1$
$7x_{1} + 2x_{2} - 19x_{3} = 1$
$6x_{1} - 13x_{3} = 6$
A: $x_{1} = 1$
$x_{2} = -3$
$x_{3} = 0$

Q: Solve the following linear system:
$x_{1} - x_{2} = 1$
$-x_{1} - x_{2} = 5$
$-x_{1} - x_{2} - 4x_{3} = 1$
A: $x_{1} = -2$
$x_{2} = -3$
$x_{3} = 1$

Q: Solve the following linear system:
$-x_{1} - 8x_{2} + 12x_{3} = 6$
$-8x_{2} + 9x_{3} = 8$
$x_{1} - 2x_{2} = 4$
A: $x_{1} = 2$
$x_{2} = -1$
$x_{3} = 0$

Q: Solve the following linear system:
$-x_{1} + 2x_{2} + 2x_{3} = 4$
$2x_{1} + x_{2} - 2x_{3} = 4$
$-x_{1} + x_{2} + x_{3} = 1$
A: $x_{1} = 2$
$x_{2} = 2$
$x_{3} = 1$

Q: Solve the following linear system:
$x_{1} + x_{3} = 2$
$5x_{1} + x_{2} + 7x_{3} = 7$
A: $x_{1} = 2 -s_1$
$x_{2} = -3 -2s_1$
$x_{3} = s_1$
where $s_1$ is a free variable

Q: Solve the following linear system:
$2x_{1} - 11x_{2} - 7x_{3} = 16$
$-x_{1} + 5x_{2} + 3x_{3} = -7$
A: $x_{1} = -3 -2s_1$
$x_{2} = -2 -s_1$
$x_{3} = s_1$
where $s_1$ is a free variable

Q: Solve the following linear system:
$-4x_{1} + 9x_{2} - 14x_{3} = -13$
$-4x_{1} + 11x_{2} - 18x_{3} = -15$
A: $x_{1} = 1 + s_1$
$x_{2} = -1 + 2s_1$
$x_{3} = s_1$
where $s_1$ is a free variable

Q: Solve the following linear system:
$x_{1} + 2x_{3} = -2$
$2x_{1} - x_{2} + 2x_{3} = -5$
A: $x_{1} = -2 -2s_1$
$x_{2} = 1 -2s_1$
$x_{3} = s_1$
where $s_1$ is a free variable

Q: Solve the following linear system:
$x_{1} + 7x_{2} + 5x_{3} = -5$
$x_{1} + 3x_{2} + x_{3} = -1$
A: $x_{1} = 2 + 2s_1$
$x_{2} = -1 -s_1$
$x_{3} = s_1$
where $s_1$ is a free variable

Q: Solve the following linear system:
$-x_{2} + 2x_{3} - x_{4} = -3$
$-2x_{1} - x_{2} + 2x_{3} - x_{4} = -3$
A: $x_{1} = 0$
$x_{2} = 3 + 2s_1 -s_2$
$x_{3} = s_1$
$x_{4} = s_2$
where $s_1$, $s_2$ are free variables

Q: Solve the following linear system:
$x_{2} - x_{4} = 0$
$-x_{1} + 3x_{2} + x_{3} - 2x_{4} = -3$
A: $x_{1} = 3 + s_1 + s_2$
$x_{2} = s_2$
$x_{3} = s_1$
$x_{4} = s_2$
where $s_1$, $s_2$ are free variables

Q: Solve the following linear system:
$5x_{1} - 2x_{2} - 9x_{3} - 6x_{4} = 5$
$3x_{1} - 2x_{2} - 7x_{3} - 2x_{4} = 3$
A: $x_{1} = 1 + s_1 + 2s_2$
$x_{2} = -2s_1 + 2s_2$
$x_{3} = s_1$
$x_{4} = s_2$
where $s_1$, $s_2$ are free variables

Q: Solve the following linear system:
$-x_{1} - 2x_{2} + 2x_{3} - 6x_{4} = 0$
$-x_{1} - 2x_{3} - 2x_{4} = 2$
A: $x_{1} = -2 -2s_1 -2s_2$
$x_{2} = 1 + 2s_1 -2s_2$
$x_{3} = s_1$
$x_{4} = s_2$
where $s_1$, $s_2$ are free variables

Q: Solve the following linear system:
$x_{2} - x_{3} + 2x_{4} = 1$
$4x_{1} + x_{2} - 9x_{3} - 2x_{4} = -7$
A: $x_{1} = -2 + 2s_1 + s_2$
$x_{2} = 1 + s_1 -2s_2$
$x_{3} = s_1$
$x_{4} = s_2$
where $s_1$, $s_2$ are free variables

Q: Solve the following linear system:
$-2x_{1} - x_{2} + 4x_{3} + 3x_{4} = 0$
$-3x_{1} - x_{2} + 5x_{3} + 5x_{4} = 1$
A: $x_{1} = -1 + s_1 + 2s_2$
$x_{2} = 2 + 2s_1 -s_2$
$x_{3} = s_1$
$x_{4} = s_2$
where $s_1$, $s_2$ are free variables

Q: Solve the following linear system:
$2x_{1} - 3x_{2} - 2x_{3} + 4x_{4} = -2$
$-4x_{1} + 3x_{2} - 2x_{3} - 2x_{4} = 4$
A: $x_{1} = -1 -2s_1 + s_2$
$x_{2} = -2s_1 + 2s_2$
$x_{3} = s_1$
$x_{4} = s_2$
where $s_1$, $s_2$ are free variables

Q: Solve the following linear system:
$6x_{1} + x_{3} + 12x_{4} = 7$
$-2x_{2} + 5x_{3} - 4x_{4} = 3$
$2x_{1} + x_{2} - 2x_{3} + 6x_{4} = 1$
A: $x_{1} = 1 -2s_1$
$x_{2} = 1 -2s_1$
$x_{3} = 1$
$x_{4} = s_1$
where $s_1$ is a free variable

Q: Solve the following linear system:
$x_{1} - x_{3} - x_{4} = 6$
$-x_{1} - x_{2} + 2x_{4} = -2$
$-x_{3} - x_{4} = 3$
A: $x_{1} = 3$
$x_{2} = -1 + 2s_1$
$x_{3} = -3 -s_1$
$x_{4} = s_1$
where $s_1$ is a free variable

Q: Solve the following linear system:
$3x_{1} + 4x_{3} - 5x_{4} = -14$
$x_{1} + x_{3} - x_{4} = -4$
$x_{2} - 2x_{4} = -1$
A: $x_{1} = -2 -s_1$
$x_{2} = -1 + 2s_1$
$x_{3} = -2 + 2s_1$
$x_{4} = s_1$
where $s_1$ is a free variable

Q: Solve the following linear system:
$2x_{1} - 3x_{3} + 5x_{4} = 12$
$x_{1} + x_{2} = 0$
$-x_{1} + x_{3} - 2x_{4} = -5$
A: $x_{1} = 3 -s_1$
$x_{2} = -3 + s_1$
$x_{3} = -2 + s_1$
$x_{4} = s_1$
where $s_1$ is a free variable

Q: Solve the following linear system:
$-x_{1} + 2x_{2} = 4$
$-x_{1} + 2x_{2} = 6$
A: This system has no solution (inconsistent).
When we solve the system, we get a contradiction of the form $0 = 1$, which is impossible.

Q: Solve the following linear system:
$7x_{1} - 2x_{2} = -12$
$7x_{1} - 2x_{2} = -9$
A: This system has no solution (inconsistent).
When we solve the system, we get a contradiction of the form $0 = 1$, which is impossible.

Q: Solve the following linear system:
$2x_{1} - x_{2} = 5$
$2x_{1} - x_{2} = 6$
A: This system has no solution (inconsistent).
When we solve the system, we get a contradiction of the form $0 = 1$, which is impossible.

Q: Solve the following linear system:
$2x_{1} - 3x_{2} = 13$
$2x_{1} - 3x_{2} = 16$
A: This system has no solution (inconsistent).
When we solve the system, we get a contradiction of the form $0 = 1$, which is impossible.

Q: Solve the following linear system:
$x_{1} - x_{2} = -4$
$x_{1} - x_{2} = -1$
A: This system has no solution (inconsistent).
When we solve the system, we get a contradiction of the form $0 = 1$, which is impossible.

Q: Solve the following linear system:
$x_{1} + 4x_{2} = -8$
$x_{1} + 4x_{2} = -6$
A: This system has no solution (inconsistent).
When we solve the system, we get a contradiction of the form $0 = 1$, which is impossible.

Q: Solve the following linear system:
$5x_{1} + x_{2} = 7$
$-5x_{1} - x_{2} = -5$
A: This system has no solution (inconsistent).
When we solve the system, we get a contradiction of the form $0 = 1$, which is impossible.

Q: Solve the following linear system:
$-5x_{1} + 6x_{2} = -16$
$5x_{1} - 6x_{2} = 17$
A: This system has no solution (inconsistent).
When we solve the system, we get a contradiction of the form $0 = 1$, which is impossible.

Q: Solve the following linear system:
$x_{1} + 4x_{2} = -9$
$x_{1} + 4x_{2} = -6$
A: This system has no solution (inconsistent).
When we solve the system, we get a contradiction of the form $0 = 1$, which is impossible.

Q: Solve the following linear system:
$2x_{1} + 5x_{2} = 20$
$-2x_{1} - 5x_{2} = -17$
A: This system has no solution (inconsistent).
When we solve the system, we get a contradiction of the form $0 = 1$, which is impossible.

Q: Solve the following linear system:
$3x_{1} - x_{2} - 3x_{3} = -9$
$-5x_{1} + x_{2} + 4x_{3} = 13$
$-2x_{1} + x_{3} = 5$
A: This system has no solution (inconsistent).
When we solve the system, we get a contradiction of the form $0 = 1$, which is impossible.

Q: Solve the following linear system:
$x_{1} + 4x_{3} = 3$
$5x_{1} - x_{2} + 14x_{3} = 12$
$6x_{1} - x_{2} + 18x_{3} = 17$
A: This system has no solution (inconsistent).
When we solve the system, we get a contradiction of the form $0 = 1$, which is impossible.

Q: Solve the following linear system:
$4x_{1} + 6x_{2} + x_{3} = -9$
$x_{1} + 2x_{2} - 2x_{3} = -3$
$-5x_{1} - 8x_{2} + x_{3} = 15$
A: This system has no solution (inconsistent).
When we solve the system, we get a contradiction of the form $0 = 1$, which is impossible.

Q: Solve the following linear system:
$x_{1} + 4x_{3} = 5$
$2x_{1} - x_{2} - 2x_{3} = -1$
$x_{1} - x_{2} - 6x_{3} = -3$
A: This system has no solution (inconsistent).
When we solve the system, we get a contradiction of the form $0 = 1$, which is impossible.

Q: Solve the following linear system:
$3x_{1} - 10x_{2} + 5x_{3} = -12$
$2x_{2} - x_{3} = 4$
$3x_{1} - 8x_{2} + 4x_{3} = -7$
A: This system has no solution (inconsistent).
When we solve the system, we get a contradiction of the form $0 = 1$, which is impossible.