Solving Systems of Equations with Matrices Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

medium
Solve using the inverse matrix: {2x+y=5x−y=1\begin{cases} 2x + y = 5 \\ x - y = 1 \end{cases}

Solution

  1. 1
    Step 1: Write as Ax=bAx = b: [211−1][xy]=[51]\begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 1 \end{bmatrix}.
  2. 2
    Step 2: det⁡(A)=−2−1=−3\det(A) = -2 - 1 = -3. A−1=1−3[−1−1−12]=[1/31/31/3−2/3]A^{-1} = \frac{1}{-3}\begin{bmatrix} -1 & -1 \\ -1 & 2 \end{bmatrix} = \begin{bmatrix} 1/3 & 1/3 \\ 1/3 & -2/3 \end{bmatrix}.
  3. 3
    Step 3: x=A−1b=[1/31/31/3−2/3][51]=[21]x = A^{-1}b = \begin{bmatrix} 1/3 & 1/3 \\ 1/3 & -2/3 \end{bmatrix}\begin{bmatrix} 5 \\ 1 \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}.
  4. 4
    Check: 2(2)+1=52(2) + 1 = 5 ✓ and 2−1=12 - 1 = 1 ✓

Answer

x=2,y=1x = 2, y = 1
Any system Ax=bAx = b with invertible AA can be solved as x=A−1bx = A^{-1}b. This matrix method is equivalent to elimination but becomes more systematic for larger systems.

About Solving Systems of Equations with Matrices

Systems of linear equations can be represented as the matrix equation Ax=bAx = b and solved using augmented matrices with row reduction (Gaussian elimination), matrix inverses (x=A−1bx = A^{-1}b), or Cramer's rule (using determinants).

Learn more about Solving Systems of Equations with Matrices →

More Solving Systems of Equations with Matrices Examples

Example 2 hard

Use Cramer's rule to solve: [formula]

Example 3 easy

Write the system [formula] as a matrix equation [formula].

Example 4 medium

Solve using the inverse: [formula]

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