Solving Systems of Equations with Matrices Math Example 4

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

Example 4

medium
Solve using the inverse: {x+2y=83x+5y=19\begin{cases} x + 2y = 8 \\ 3x + 5y = 19 \end{cases}

Solution

  1. 1
    det⁡=5−6=−1\det = 5 - 6 = -1. A−1=1−1[5−2−31]=[−523−1]A^{-1} = \frac{1}{-1}\begin{bmatrix} 5 & -2 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix} -5 & 2 \\ 3 & -1 \end{bmatrix}.
  2. 2
    [xy]=[−523−1][819]=[−25]\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} -5 & 2 \\ 3 & -1 \end{bmatrix}\begin{bmatrix} 8 \\ 19 \end{bmatrix} = \begin{bmatrix} -2 \\ 5 \end{bmatrix}. So x=−2,y=5x = -2, y = 5.

Answer

x=−2,y=5x = -2, y = 5
Compute A−1A^{-1}, then multiply A−1bA^{-1}b to get the solution vector. Verify: −2+10=8-2 + 10 = 8 ✓ and −6+25=19-6 + 25 = 19 ✓.

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 1 medium

Solve using the inverse matrix: [formula]

Example 2 hard

Use Cramer's rule to solve: [formula]

Example 3 easy

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

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