Solve Using an Inverse Matrix -2x+5y=3 8x-2y=6

$- 2 x + 5 y = 3$ $8 x - 2 y = 6$

Step 1

Find the $A X = B$ from the system of equations.

$[\begin{array}{c} - 2 & 5 \\ 8 & - 2 \end{array}] \cdot [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 3 \\ 6 \end{array}]$

Step 2

Find the inverse of the coefficient matrix.

The inverse of a $2 \times 2$ matrix can be found using the formula $\frac{1}{a d - b c} [\begin{array}{c} d & - b \\ - c & a \end{array}]$ where $a d - b c$ is the determinant.

Find the determinant.

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- 2 \cdot - 2 - 8 \cdot 5$

Simplify the determinant.

Simplify each term.

Multiply $- 2$ by $- 2$ .

$4 - 8 \cdot 5$

Multiply $- 8$ by $5$ .

$4 - 40$

Subtract $40$ from $4$ .

$- 36$

Substitute the known values into the formula for the inverse.

$\frac{1}{- 36} [\begin{array}{c} - 2 & - 5 \\ - 8 & - 2 \end{array}]$

Move the negative in front of the fraction.

$- \frac{1}{36} [\begin{array}{c} - 2 & - 5 \\ - 8 & - 2 \end{array}]$

Multiply $- \frac{1}{36}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{36} \cdot - 2 & - \frac{1}{36} \cdot - 5 \\ - \frac{1}{36} \cdot - 8 & - \frac{1}{36} \cdot - 2 \end{array}]$

Simplify each element in the matrix.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{36}$ into the numerator.

$[\begin{array}{c} \frac{- 1}{36} \cdot - 2 & - \frac{1}{36} \cdot - 5 \\ - \frac{1}{36} \cdot - 8 & - \frac{1}{36} \cdot - 2 \end{array}]$

Factor $2$ out of $36$ .

$[\begin{array}{c} \frac{- 1}{2 (18)} \cdot - 2 & - \frac{1}{36} \cdot - 5 \\ - \frac{1}{36} \cdot - 8 & - \frac{1}{36} \cdot - 2 \end{array}]$

Factor $2$ out of $- 2$ .

$[\begin{array}{c} \frac{- 1}{2 \cdot 18} \cdot (2 \cdot - 1) & - \frac{1}{36} \cdot - 5 \\ - \frac{1}{36} \cdot - 8 & - \frac{1}{36} \cdot - 2 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{- 1}{2 \cdot 18} \cdot (2 \cdot - 1) & - \frac{1}{36} \cdot - 5 \\ - \frac{1}{36} \cdot - 8 & - \frac{1}{36} \cdot - 2 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{- 1}{18} \cdot - 1 & - \frac{1}{36} \cdot - 5 \\ - \frac{1}{36} \cdot - 8 & - \frac{1}{36} \cdot - 2 \end{array}]$

Combine $\frac{- 1}{18}$ and $- 1$ .

$[\begin{array}{c} \frac{- - 1}{18} & - \frac{1}{36} \cdot - 5 \\ - \frac{1}{36} \cdot - 8 & - \frac{1}{36} \cdot - 2 \end{array}]$

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} \frac{1}{18} & - \frac{1}{36} \cdot - 5 \\ - \frac{1}{36} \cdot - 8 & - \frac{1}{36} \cdot - 2 \end{array}]$

Multiply $- \frac{1}{36} \cdot - 5$ .

Multiply $- 5$ by $- 1$ .

$[\begin{array}{c} \frac{1}{18} & 5 (\frac{1}{36}) \\ - \frac{1}{36} \cdot - 8 & - \frac{1}{36} \cdot - 2 \end{array}]$

Combine $5$ and $\frac{1}{36}$ .

$[\begin{array}{c} \frac{1}{18} & \frac{5}{36} \\ - \frac{1}{36} \cdot - 8 & - \frac{1}{36} \cdot - 2 \end{array}]$

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{1}{36}$ into the numerator.

$[\begin{array}{c} \frac{1}{18} & \frac{5}{36} \\ \frac{- 1}{36} \cdot - 8 & - \frac{1}{36} \cdot - 2 \end{array}]$

Factor $4$ out of $36$ .

$[\begin{array}{c} \frac{1}{18} & \frac{5}{36} \\ \frac{- 1}{4 (9)} \cdot - 8 & - \frac{1}{36} \cdot - 2 \end{array}]$

Factor $4$ out of $- 8$ .

$[\begin{array}{c} \frac{1}{18} & \frac{5}{36} \\ \frac{- 1}{4 \cdot 9} \cdot (4 \cdot - 2) & - \frac{1}{36} \cdot - 2 \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{1}{18} & \frac{5}{36} \\ \frac{- 1}{4 \cdot 9} \cdot (4 \cdot - 2) & - \frac{1}{36} \cdot - 2 \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{1}{18} & \frac{5}{36} \\ \frac{- 1}{9} \cdot - 2 & - \frac{1}{36} \cdot - 2 \end{array}]$

Combine $\frac{- 1}{9}$ and $- 2$ .

$[\begin{array}{c} \frac{1}{18} & \frac{5}{36} \\ \frac{- - 2}{9} & - \frac{1}{36} \cdot - 2 \end{array}]$

Multiply $- 1$ by $- 2$ .

$[\begin{array}{c} \frac{1}{18} & \frac{5}{36} \\ \frac{2}{9} & - \frac{1}{36} \cdot - 2 \end{array}]$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{36}$ into the numerator.

$[\begin{array}{c} \frac{1}{18} & \frac{5}{36} \\ \frac{2}{9} & \frac{- 1}{36} \cdot - 2 \end{array}]$

Factor $2$ out of $36$ .

$[\begin{array}{c} \frac{1}{18} & \frac{5}{36} \\ \frac{2}{9} & \frac{- 1}{2 (18)} \cdot - 2 \end{array}]$

Factor $2$ out of $- 2$ .

$[\begin{array}{c} \frac{1}{18} & \frac{5}{36} \\ \frac{2}{9} & \frac{- 1}{2 \cdot 18} \cdot (2 \cdot - 1) \end{array}]$

Cancel the common factor.

$[\begin{array}{c} \frac{1}{18} & \frac{5}{36} \\ \frac{2}{9} & \frac{- 1}{2 \cdot 18} \cdot (2 \cdot - 1) \end{array}]$

Rewrite the expression.

$[\begin{array}{c} \frac{1}{18} & \frac{5}{36} \\ \frac{2}{9} & \frac{- 1}{18} \cdot - 1 \end{array}]$

Combine $\frac{- 1}{18}$ and $- 1$ .

$[\begin{array}{c} \frac{1}{18} & \frac{5}{36} \\ \frac{2}{9} & \frac{- - 1}{18} \end{array}]$

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} \frac{1}{18} & \frac{5}{36} \\ \frac{2}{9} & \frac{1}{18} \end{array}]$

Step 3

Left multiply both sides of the matrix equation by the inverse matrix.

$([\begin{array}{c} \frac{1}{18} & \frac{5}{36} \\ \frac{2}{9} & \frac{1}{18} \end{array}] \cdot [\begin{array}{c} - 2 & 5 \\ 8 & - 2 \end{array}]) \cdot [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} \frac{1}{18} & \frac{5}{36} \\ \frac{2}{9} & \frac{1}{18} \end{array}] \cdot [\begin{array}{c} 3 \\ 6 \end{array}]$

Step 4

Any matrix multiplied by its inverse is equal to $1$ all the time. $A \cdot A^{- 1} = 1$ .

$[\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} \frac{1}{18} & \frac{5}{36} \\ \frac{2}{9} & \frac{1}{18} \end{array}] \cdot [\begin{array}{c} 3 \\ 6 \end{array}]$

Step 5

Multiply $[\begin{array}{c} \frac{1}{18} & \frac{5}{36} \\ \frac{2}{9} & \frac{1}{18} \end{array}] [\begin{array}{c} 3 \\ 6 \end{array}]$ .

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} \frac{1}{18} \cdot 3 + \frac{5}{36} \cdot 6 \\ \frac{2}{9} \cdot 3 + \frac{1}{18} \cdot 6 \end{array}]$

Simplify each element of the matrix by multiplying out all the expressions.

$[\begin{array}{c} 1 \\ 1 \end{array}]$

Step 6

Simplify the left and right side.

$[\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 1 \\ 1 \end{array}]$

Step 7

Find the solution.

$x = 1$

$y = 1$

Solve Using an Inverse Matrix -2x+5y=3 8x-2y=6

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