Solving Systems of Equations with Matrices Math Example 2

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Example 2

hard
Use Cramer's rule to solve: {3x+2y=7xโˆ’y=1\begin{cases} 3x + 2y = 7 \\ x - y = 1 \end{cases}

Solution

  1. 1
    Step 1: detโก(A)=detโก[321โˆ’1]=โˆ’3โˆ’2=โˆ’5\det(A) = \det\begin{bmatrix} 3 & 2 \\ 1 & -1 \end{bmatrix} = -3 - 2 = -5.
  2. 2
    Step 2: x=detโก[721โˆ’1]โˆ’5=โˆ’7โˆ’2โˆ’5=โˆ’9โˆ’5=95x = \frac{\det\begin{bmatrix} 7 & 2 \\ 1 & -1 \end{bmatrix}}{-5} = \frac{-7-2}{-5} = \frac{-9}{-5} = \frac{9}{5}.
  3. 3
    Step 3: y=detโก[3711]โˆ’5=3โˆ’7โˆ’5=โˆ’4โˆ’5=45y = \frac{\det\begin{bmatrix} 3 & 7 \\ 1 & 1 \end{bmatrix}}{-5} = \frac{3-7}{-5} = \frac{-4}{-5} = \frac{4}{5}.
  4. 4
    Check: 3(9/5)+2(4/5)=27/5+8/5=35/5=73(9/5) + 2(4/5) = 27/5 + 8/5 = 35/5 = 7 โœ“

Answer

x=95,y=45x = \frac{9}{5}, y = \frac{4}{5}
Cramer's rule finds each variable by replacing its column in AA with the constant vector bb and dividing the resulting determinant by detโก(A)\det(A). It requires detโก(A)โ‰ 0\det(A) \neq 0.

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=Aโˆ’1bx = A^{-1}b), or Cramer's rule (using determinants).

Learn more about Solving Systems of Equations with Matrices โ†’

More Solving Systems of Equations with Matrices Examples

Example 1 medium

Solve using the inverse matrix: [formula]

Example 3 easy

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

Example 4 medium

Solve using the inverse: [formula]

โ† All Solving Systems of Equations with Matrices Examples Solving Systems of Equations with Matrices Concept