Practice Solving Systems of Equations with Matrices in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
Systems of linear equations can be represented as the matrix equation Ax = b and solved using augmented matrices with row reduction (Gaussian elimination), matrix inverses (x = A^{-1}b), or Cramer's rule (using determinants).
Instead of juggling multiple equations with substitution or elimination, pack everything into a matrix and use systematic row operations. It is like organizing a messy desk—once the equations are neatly arranged in a matrix, a mechanical process (row reduction) reveals the answer. Each row operation is an allowed algebraic move (swap equations, scale an equation, add equations) performed on the matrix.
Example 1
mediumSolve using the inverse matrix: \begin{cases} 2x + y = 5 \\ x - y = 1 \end{cases}
Example 2
hardUse Cramer's rule to solve: \begin{cases} 3x + 2y = 7 \\ x - y = 1 \end{cases}
Example 3
easyWrite the system \begin{cases} x + 3y = 7 \\ 2x - y = 4 \end{cases} as a matrix equation Ax = b.
Example 4
mediumSolve using the inverse: \begin{cases} x + 2y = 8 \\ 3x + 5y = 19 \end{cases}