Solve Using an Augmented Matrix 8x+5y+u=7 , x+6y+v=9 , -32x-2y+m=0

$8 x + 5 y + u = 7$ , $x + 6 y + v = 9$ , $- 32 x - 2 y + m = 0$

Write the system of equations in matrix form.

$[\begin{array}{c} 0 & 1 & 0 & 8 & 5 & 7 \\ 0 & 0 & 1 & 1 & 6 & 9 \\ 1 & 0 & 0 & - 32 & - 2 & 0 \end{array}]$

Row reduce.

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Exchange row $3$ and row $1$ to organize the zeros into position.

$[\begin{array}{c} 1 & 0 & 0 & - 32 & - 2 & 0 \\ 0 & 0 & 1 & 1 & 6 & 9 \\ 0 & 1 & 0 & 8 & 5 & 7 \end{array}]$

$R_{3} \leftrightarrow R$

Exchange row $3$ and row $2$ to organize the zeros into position.

$[\begin{array}{c} 1 & 0 & 0 & - 32 & - 2 & 0 \\ 0 & 1 & 0 & 8 & 5 & 7 \\ 0 & 0 & 1 & 1 & 6 & 9 \end{array}]$

$R_{3} \leftrightarrow R$

$[\begin{array}{c} 1 & 0 & 0 & - 32 & - 2 & 0 \\ 0 & 1 & 0 & 8 & 5 & 7 \\ 0 & 0 & 1 & 1 & 6 & 9 \end{array}]$

$R_{3} \leftrightarrow R$

Use the result matrix to declare the final solutions to the system of equations.

$m - 32 x - 2 y = 0$

$u + 8 x + 5 y = 7$

$v + x + 6 y = 9$

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

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Add $32 x$ to both sides of the equation.

$m - 2 y = 32 x$

$u + 8 x + 5 y = 7$

$v + x + 6 y = 9$

Add $2 y$ to both sides of the equation.

$m = 32 x + 2 y$

$u + 8 x + 5 y = 7$

$v + x + 6 y = 9$

$m = 32 x + 2 y$

$u + 8 x + 5 y = 7$

$v + x + 6 y = 9$

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

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Subtract $8 x$ from both sides of the equation.

$m = 32 x + 2 y$

$u + 5 y = 7 - 8 x$

$v + x + 6 y = 9$

Subtract $5 y$ from both sides of the equation.

$m = 32 x + 2 y$

$u = 7 - 8 x - 5 y$

$v + x + 6 y = 9$

$m = 32 x + 2 y$

$u = 7 - 8 x - 5 y$

$v + x + 6 y = 9$

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

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Subtract $x$ from both sides of the equation.

$m = 32 x + 2 y$

$u = 7 - 8 x - 5 y$

$v + 6 y = 9 - x$

Subtract $6 y$ from both sides of the equation.

$m = 32 x + 2 y$

$u = 7 - 8 x - 5 y$

$v = 9 - x - 6 y$

$m = 32 x + 2 y$

$u = 7 - 8 x - 5 y$

$v = 9 - x - 6 y$

The solution is the set of ordered pairs that makes the system true.

$(32 x + 2 y, 7 - 8 x - 5 y, 9 - x - 6 y, x, y)$

Solve Using an Augmented Matrix 8x+5y+u=7 , x+6y+v=9 , -32x-2y+m=0

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