Linear Systems — Practice Problems and Worked Solutions

17 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Solve the linear system for the variable vector.

   $$\left[\begin{array}{cc}1& 1\\ 2& -1\end{array}\right]x=\left[\begin{array}{c}5\\ 1\end{array}\right]$$

   Answer: $\left[\begin{array}{c}2\\ 3\end{array}\right]$

   Step-by-step solution

   1. $$x=A^{-1}b$$
   2. $$x=\left[\begin{array}{c}2\\ 3\end{array}\right]$$
   3. $$Ax=\left[\begin{array}{c}5\\ 1\end{array}\right]=\left[\begin{array}{c}5\\ 1\end{array}\right]$$
2. 2.

   Solve the linear system for the variable vector.

   $$\left[\begin{array}{cc}2& 1\\ 1& -1\end{array}\right]x=\left[\begin{array}{c}7\\ 2\end{array}\right]$$

   Answer: $\left[\begin{array}{c}3\\ 1\end{array}\right]$

   Step-by-step solution

   1. $$x=A^{-1}b$$
   2. $$x=\left[\begin{array}{c}3\\ 1\end{array}\right]$$
   3. $$Ax=\left[\begin{array}{c}7\\ 2\end{array}\right]=\left[\begin{array}{c}7\\ 2\end{array}\right]$$
3. 3.

   Solve the linear system for the variable vector.

   $$\left[\begin{array}{cc}3& -1\\ 1& 2\end{array}\right]x=\left[\begin{array}{c}4\\ 5\end{array}\right]$$

   Answer: $\left[\begin{array}{c}\frac{13}{7}\\ \frac{11}{7}\end{array}\right]$

   Step-by-step solution

   1. $$x=A^{-1}b$$
   2. $$x=\left[\begin{array}{c}\frac{13}{7}\\ \frac{11}{7}\end{array}\right]$$
   3. $$Ax=\left[\begin{array}{c}4\\ 5\end{array}\right]=\left[\begin{array}{c}4\\ 5\end{array}\right]$$
4. 4.

   Solve the linear system for the variable vector.

   $$\left[\begin{array}{cc}1& 2\\ 3& -1\end{array}\right]x=\left[\begin{array}{c}8\\ 1\end{array}\right]$$

   Answer: $\left[\begin{array}{c}\frac{10}{7}\\ \frac{23}{7}\end{array}\right]$

   Step-by-step solution

   1. $$x=A^{-1}b$$
   2. $$x=\left[\begin{array}{c}\frac{10}{7}\\ \frac{23}{7}\end{array}\right]$$
   3. $$Ax=\left[\begin{array}{c}8\\ 1\end{array}\right]=\left[\begin{array}{c}8\\ 1\end{array}\right]$$
5. 5.

   Solve the linear system for the variable vector.

   $$\left[\begin{array}{cc}2& -3\\ 4& 1\end{array}\right]x=\left[\begin{array}{c}-1\\ 9\end{array}\right]$$

   Answer: $\left[\begin{array}{c}\frac{13}{7}\\ \frac{11}{7}\end{array}\right]$

   Step-by-step solution

   1. $$x=A^{-1}b$$
   2. $$x=\left[\begin{array}{c}\frac{13}{7}\\ \frac{11}{7}\end{array}\right]$$
   3. $$Ax=\left[\begin{array}{c}-1\\ 9\end{array}\right]=\left[\begin{array}{c}-1\\ 9\end{array}\right]$$
6. 6.

   Solve the linear system for the variable vector.

   $$\left[\begin{array}{cc}5& 2\\ -1& 3\end{array}\right]x=\left[\begin{array}{c}12\\ 4\end{array}\right]$$

   Answer: $\left[\begin{array}{c}\frac{28}{17}\\ \frac{32}{17}\end{array}\right]$

   Step-by-step solution

   1. $$x=A^{-1}b$$
   2. $$x=\left[\begin{array}{c}\frac{28}{17}\\ \frac{32}{17}\end{array}\right]$$
   3. $$Ax=\left[\begin{array}{c}12\\ 4\end{array}\right]=\left[\begin{array}{c}12\\ 4\end{array}\right]$$
7. 7.

   Solve the linear system for the variable vector.

   $$\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]x=\left[\begin{array}{c}6\\ 2\end{array}\right]$$

   Answer: $\left[\begin{array}{c}4\\ 2\end{array}\right]$

   Step-by-step solution

   1. $$x=A^{-1}b$$
   2. $$x=\left[\begin{array}{c}4\\ 2\end{array}\right]$$
   3. $$Ax=\left[\begin{array}{c}6\\ 2\end{array}\right]=\left[\begin{array}{c}6\\ 2\end{array}\right]$$
8. 8.

   Solve the linear system for the variable vector.

   $$\left[\begin{array}{cc}3& 1\\ 2& -2\end{array}\right]x=\left[\begin{array}{c}7\\ 2\end{array}\right]$$

   Answer: $\left[\begin{array}{c}2\\ 1\end{array}\right]$

   Step-by-step solution

   1. $$x=A^{-1}b$$
   2. $$x=\left[\begin{array}{c}2\\ 1\end{array}\right]$$
   3. $$Ax=\left[\begin{array}{c}7\\ 2\end{array}\right]=\left[\begin{array}{c}7\\ 2\end{array}\right]$$
9. 9.

   Solve the linear system for the variable vector.

   $$\left[\begin{array}{cc}2& 5\\ 1& 3\end{array}\right]x=\left[\begin{array}{c}11\\ 6\end{array}\right]$$

   Answer: $\left[\begin{array}{c}3\\ 1\end{array}\right]$

   Step-by-step solution

   1. $$x=A^{-1}b$$
   2. $$x=\left[\begin{array}{c}3\\ 1\end{array}\right]$$
   3. $$Ax=\left[\begin{array}{c}11\\ 6\end{array}\right]=\left[\begin{array}{c}11\\ 6\end{array}\right]$$
10. 10.

    Solve the linear system for the variable vector.

    $$\left[\begin{array}{cc}4& -1\\ 2& 3\end{array}\right]x=\left[\begin{array}{c}10\\ 12\end{array}\right]$$

    Answer: $\left[\begin{array}{c}3\\ 2\end{array}\right]$

    Step-by-step solution

    1. $$x=A^{-1}b$$
    2. $$x=\left[\begin{array}{c}3\\ 2\end{array}\right]$$
    3. $$Ax=\left[\begin{array}{c}10\\ 12\end{array}\right]=\left[\begin{array}{c}10\\ 12\end{array}\right]$$
11. 11.

    Solve the linear system for the variable vector.

    $$\left[\begin{array}{cc}1& 4\\ 2& 3\end{array}\right]x=\left[\begin{array}{c}9\\ 8\end{array}\right]$$

    Answer: $\left[\begin{array}{c}1\\ 2\end{array}\right]$

    Step-by-step solution

    1. $$x=A^{-1}b$$
    2. $$x=\left[\begin{array}{c}1\\ 2\end{array}\right]$$
    3. $$Ax=\left[\begin{array}{c}9\\ 8\end{array}\right]=\left[\begin{array}{c}9\\ 8\end{array}\right]$$
12. 12.

    Solve the linear system for the variable vector.

    $$\left[\begin{array}{cc}3& 2\\ 1& -1\end{array}\right]x=\left[\begin{array}{c}13\\ 1\end{array}\right]$$

    Answer: $\left[\begin{array}{c}3\\ 2\end{array}\right]$

    Step-by-step solution

    1. $$x=A^{-1}b$$
    2. $$x=\left[\begin{array}{c}3\\ 2\end{array}\right]$$
    3. $$Ax=\left[\begin{array}{c}13\\ 1\end{array}\right]=\left[\begin{array}{c}13\\ 1\end{array}\right]$$
13. 13.

    Solve the linear system for the variable vector.

    $$\left[\begin{array}{cc}2& 1\\ 5& 3\end{array}\right]x=\left[\begin{array}{c}8\\ 21\end{array}\right]$$

    Answer: $\left[\begin{array}{c}3\\ 2\end{array}\right]$

    Step-by-step solution

    1. $$x=A^{-1}b$$
    2. $$x=\left[\begin{array}{c}3\\ 2\end{array}\right]$$
    3. $$Ax=\left[\begin{array}{c}8\\ 21\end{array}\right]=\left[\begin{array}{c}8\\ 21\end{array}\right]$$
14. 14.

    Solve the linear system for the variable vector.

    $$\left[\begin{array}{cc}1& -2\\ 3& 4\end{array}\right]x=\left[\begin{array}{c}-3\\ 19\end{array}\right]$$

    Answer: $\left[\begin{array}{c}\frac{13}{5}\\ \frac{14}{5}\end{array}\right]$

    Step-by-step solution

    1. $$x=A^{-1}b$$
    2. $$x=\left[\begin{array}{c}\frac{13}{5}\\ \frac{14}{5}\end{array}\right]$$
    3. $$Ax=\left[\begin{array}{c}-3\\ 19\end{array}\right]=\left[\begin{array}{c}-3\\ 19\end{array}\right]$$
15. 15.

    Solve the linear system for the variable vector.

    $$\left[\begin{array}{cc}6& 1\\ 2& 5\end{array}\right]x=\left[\begin{array}{c}20\\ 16\end{array}\right]$$

    Answer: $\left[\begin{array}{c}3\\ 2\end{array}\right]$

    Step-by-step solution

    1. $$x=A^{-1}b$$
    2. $$x=\left[\begin{array}{c}3\\ 2\end{array}\right]$$
    3. $$Ax=\left[\begin{array}{c}20\\ 16\end{array}\right]=\left[\begin{array}{c}20\\ 16\end{array}\right]$$
16. 16.

    Solve the linear system for the variable vector.

    $$\left[\begin{array}{cc}1& 3\\ 4& -2\end{array}\right]x=\left[\begin{array}{c}10\\ 12\end{array}\right]$$

    Answer: $\left[\begin{array}{c}4\\ 2\end{array}\right]$

    Step-by-step solution

    1. $$x=A^{-1}b$$
    2. $$x=\left[\begin{array}{c}4\\ 2\end{array}\right]$$
    3. $$Ax=\left[\begin{array}{c}10\\ 12\end{array}\right]=\left[\begin{array}{c}10\\ 12\end{array}\right]$$
17. 17.

    Solve the linear system for the variable vector.

    $$\left[\begin{array}{cc}2& 7\\ 1& 4\end{array}\right]x=\left[\begin{array}{c}15\\ 9\end{array}\right]$$

    Answer: $\left[\begin{array}{c}-3\\ 3\end{array}\right]$

    Step-by-step solution

    1. $$x=A^{-1}b$$
    2. $$x=\left[\begin{array}{c}-3\\ 3\end{array}\right]$$
    3. $$Ax=\left[\begin{array}{c}15\\ 9\end{array}\right]=\left[\begin{array}{c}15\\ 9\end{array}\right]$$

More from Linear Algebra

• Vector Operations
• Dot Products
• Matrix Products
• Row Reduction
• Determinants
• Matrix Inverses
• Linear Independence
• Rank & Basis
• Orthogonal Projections
• Linear Transformations
• Eigenvalues