Solving Systems of Equations with Matrices Math Example 3

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Example 3

easy
Write the system {x+3y=72x−y=4\begin{cases} x + 3y = 7 \\ 2x - y = 4 \end{cases} as a matrix equation Ax=bAx = b.

Solution

  1. 1
    A=[132−1]A = \begin{bmatrix} 1 & 3 \\ 2 & -1 \end{bmatrix}, x=[xy]x = \begin{bmatrix} x \\ y \end{bmatrix}, b=[74]b = \begin{bmatrix} 7 \\ 4 \end{bmatrix}.
  2. 2
    Matrix equation: [132−1][xy]=[74]\begin{bmatrix} 1 & 3 \\ 2 & -1 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 7 \\ 4 \end{bmatrix}.

Answer

[132−1][xy]=[74]\begin{bmatrix} 1 & 3 \\ 2 & -1 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 7 \\ 4 \end{bmatrix}
Every linear system can be written as Ax=bAx = b where AA holds the coefficients, xx is the variable vector, and bb is the constant vector. This is the first step in any matrix-based solution method.

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 4 medium

Solve using the inverse: [formula]

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