Solve the Matrix Equation [[12,-13],[17,-22]][[X],[Y]]=[[7],[-51]]

$[\begin{array}{c} 12 & - 13 \\ 17 & - 22 \end{array}] [\begin{array}{c} X \\ Y \end{array}] = [\begin{array}{c} 7 \\ - 51 \end{array}]$

Step 1

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

$[\begin{array}{c} 12 X - 13 Y \\ 17 X - 22 Y \end{array}] = [\begin{array}{c} 7 \\ - 51 \end{array}]$

Step 2

Write as a linear system of equations.

$12 X - 13 Y = 7 17 X - 22 Y = - 51$

Step 3

Solve the system of equations.

Solve for $X$ in $12 X - 13 Y = 7$ .

Add $13 Y$ to both sides of the equation.

$12 X = 7 + 13 Y$

$17 X - 22 Y = - 51$

Divide each term in $12 X = 7 + 13 Y$ by $12$ and simplify.

Divide each term in $12 X = 7 + 13 Y$ by $12$ .

$\frac{12 X}{12} = \frac{7}{12} + \frac{13 Y}{12}$

$17 X - 22 Y = - 51$

Simplify the left side.

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12 X}{12} = \frac{7}{12} + \frac{13 Y}{12}$

$17 X - 22 Y = - 51$

Divide $X$ by $1$ .

$X = \frac{7}{12} + \frac{13 Y}{12}$

$17 X - 22 Y = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$17 X - 22 Y = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$17 X - 22 Y = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$17 X - 22 Y = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$17 X - 22 Y = - 51$

Replace all occurrences of $X$ with $\frac{7}{12} + \frac{13 Y}{12}$ in each equation.

Replace all occurrences of $X$ in $17 X - 22 Y = - 51$ with $\frac{7}{12} + \frac{13 Y}{12}$ .

$17 (\frac{7}{12} + \frac{13 Y}{12}) - 22 Y = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Simplify the left side.

Simplify $17 (\frac{7}{12} + \frac{13 Y}{12}) - 22 Y$ .

Simplify each term.

Apply the distributive property.

$17 (\frac{7}{12}) + 17 (\frac{13 Y}{12}) - 22 Y = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Multiply $17 (\frac{7}{12})$ .

Combine $17$ and $\frac{7}{12}$ .

$\frac{17 \cdot 7}{12} + 17 (\frac{13 Y}{12}) - 22 Y = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Multiply $17$ by $7$ .

$\frac{119}{12} + 17 (\frac{13 Y}{12}) - 22 Y = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$\frac{119}{12} + 17 (\frac{13 Y}{12}) - 22 Y = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Multiply $17 \frac{13 Y}{12}$ .

Combine $17$ and $\frac{13 Y}{12}$ .

$\frac{119}{12} + \frac{17 (13 Y)}{12} - 22 Y = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Multiply $13$ by $17$ .

$\frac{119}{12} + \frac{221 Y}{12} - 22 Y = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$\frac{119}{12} + \frac{221 Y}{12} - 22 Y = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$\frac{119}{12} + \frac{221 Y}{12} - 22 Y = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

To write $- 22 Y$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$\frac{119}{12} + \frac{221 Y}{12} - 22 Y \cdot \frac{12}{12} = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Combine $- 22 Y$ and $\frac{12}{12}$ .

$\frac{119}{12} + \frac{221 Y}{12} + \frac{- 22 Y \cdot 12}{12} = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Combine the numerators over the common denominator.

$\frac{119}{12} + \frac{221 Y - 22 Y \cdot 12}{12} = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Combine the numerators over the common denominator.

$\frac{119 + 221 Y - 22 Y \cdot 12}{12} = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Multiply $12$ by $- 22$ .

$\frac{119 + 221 Y - 264 Y}{12} = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Subtract $264 Y$ from $221 Y$ .

$\frac{119 - 43 Y}{12} = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$\frac{119 - 43 Y}{12} = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$\frac{119 - 43 Y}{12} = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$\frac{119 - 43 Y}{12} = - 51$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Solve for $Y$ in $\frac{119 - 43 Y}{12} = - 51$ .

Multiply both sides by $12$ .

$\frac{119 - 43 Y}{12} \cdot 12 = - 51 \cdot 12$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Simplify.

Simplify the left side.

Simplify $\frac{119 - 43 Y}{12} \cdot 12$ .

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{119 - 43 Y}{12} \cdot 12 = - 51 \cdot 12$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Rewrite the expression.

$119 - 43 Y = - 51 \cdot 12$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$119 - 43 Y = - 51 \cdot 12$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Reorder $119$ and $- 43 Y$ .

$- 43 Y + 119 = - 51 \cdot 12$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$- 43 Y + 119 = - 51 \cdot 12$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$- 43 Y + 119 = - 51 \cdot 12$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Simplify the right side.

Multiply $- 51$ by $12$ .

$- 43 Y + 119 = - 612$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$- 43 Y + 119 = - 612$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$- 43 Y + 119 = - 612$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Solve for $Y$ .

Move all terms not containing $Y$ to the right side of the equation.

Subtract $119$ from both sides of the equation.

$- 43 Y = - 612 - 119$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Subtract $119$ from $- 612$ .

$- 43 Y = - 731$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$- 43 Y = - 731$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Divide each term in $- 43 Y = - 731$ by $- 43$ and simplify.

Divide each term in $- 43 Y = - 731$ by $- 43$ .

$\frac{- 43 Y}{- 43} = \frac{- 731}{- 43}$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Simplify the left side.

Cancel the common factor of $- 43$ .

Cancel the common factor.

$\frac{- 43 Y}{- 43} = \frac{- 731}{- 43}$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Divide $Y$ by $1$ .

$Y = \frac{- 731}{- 43}$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$Y = \frac{- 731}{- 43}$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$Y = \frac{- 731}{- 43}$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Simplify the right side.

Divide $- 731$ by $- 43$ .

$Y = 17$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$Y = 17$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$Y = 17$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$Y = 17$

$X = \frac{7}{12} + \frac{13 Y}{12}$

$Y = 17$

$X = \frac{7}{12} + \frac{13 Y}{12}$

Replace all occurrences of $Y$ with $17$ in each equation.

Replace all occurrences of $Y$ in $X = \frac{7}{12} + \frac{13 Y}{12}$ with $17$ .

$X = \frac{7}{12} + \frac{13 (17)}{12}$

$Y = 17$

Simplify the right side.

Simplify $\frac{7}{12} + \frac{13 (17)}{12}$ .

Combine the numerators over the common denominator.

$X = \frac{7 + 13 (17)}{12}$

$Y = 17$

Simplify the expression.

Multiply $13$ by $17$ .

$X = \frac{7 + 221}{12}$

$Y = 17$

Add $7$ and $221$ .

$X = \frac{228}{12}$

$Y = 17$

Divide $228$ by $12$ .

$X = 19$

$Y = 17$

$X = 19$

$Y = 17$

$X = 19$

$Y = 17$

$X = 19$

$Y = 17$

$X = 19$

$Y = 17$

List all of the solutions.

$X = 19, Y = 17$

Solve the Matrix Equation [[12,-13],[17,-22]][[X],[Y]]=[[7],[-51]]

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