Find the Variables [[1,2,-1],[1,3,2],[2,6,1]]*[[x],[y],[z]]=[[1],[7],[8]]

$[\begin{array}{c} 1 & 2 & - 1 \\ 1 & 3 & 2 \\ 2 & 6 & 1 \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 1 \\ 7 \\ 8 \end{array}]$

Step 1

Simplify the left side of the equation.

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

$[\begin{array}{c} 1 \cdot x + 2 \cdot y - 1 \cdot z \\ 1 \cdot x + 3 \cdot y + 2 \cdot z \\ 2 \cdot x + 6 \cdot y + 1 \cdot z \end{array}] = [\begin{array}{c} 1 \\ 7 \\ 8 \end{array}]$

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

$[\begin{array}{c} x + 2 y - z \\ x + 3 y + 2 z \\ 2 x + 6 y + z \end{array}] = [\begin{array}{c} 1 \\ 7 \\ 8 \end{array}]$

Step 2

The matrix equation can be written as a set of equations.

$x + 2 y - z = 1$

$x + 3 y + 2 z = 7$

$2 x + 6 y + z = 8$

Step 3

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

Subtract $2 y$ from both sides of the equation.

$x - z = 1 - 2 y$

$x + 3 y + 2 z = 7$

$2 x + 6 y + z = 8$

Add $z$ to both sides of the equation.

$x = 1 - 2 y + z$

$x + 3 y + 2 z = 7$

$2 x + 6 y + z = 8$

$x = 1 - 2 y + z$

$x + 3 y + 2 z = 7$

$2 x + 6 y + z = 8$

Step 4

Replace all occurrences of $x$ with $1 - 2 y + z$ in each equation.

Replace all occurrences of $x$ in $x + 3 y + 2 z = 7$ with $1 - 2 y + z$ .

$(1 - 2 y + z) + 3 y + 2 z = 7$

$x = 1 - 2 y + z$

$2 x + 6 y + z = 8$

Simplify the left side.

Simplify $(1 - 2 y + z) + 3 y + 2 z$ .

Add $- 2 y$ and $3 y$ .

$1 + y + z + 2 z = 7$

$x = 1 - 2 y + z$

$2 x + 6 y + z = 8$

Add $z$ and $2 z$ .

$1 + y + 3 z = 7$

$x = 1 - 2 y + z$

$2 x + 6 y + z = 8$

$1 + y + 3 z = 7$

$x = 1 - 2 y + z$

$2 x + 6 y + z = 8$

$1 + y + 3 z = 7$

$x = 1 - 2 y + z$

$2 x + 6 y + z = 8$

Replace all occurrences of $x$ in $2 x + 6 y + z = 8$ with $1 - 2 y + z$ .

$2 (1 - 2 y + z) + 6 y + z = 8$

$1 + y + 3 z = 7$

$x = 1 - 2 y + z$

Simplify the left side.

Simplify $2 (1 - 2 y + z) + 6 y + z$ .

Simplify each term.

Apply the distributive property.

$2 \cdot 1 + 2 (- 2 y) + 2 z + 6 y + z = 8$

$1 + y + 3 z = 7$

$x = 1 - 2 y + z$

Simplify.

Multiply $2$ by $1$ .

$2 + 2 (- 2 y) + 2 z + 6 y + z = 8$

$1 + y + 3 z = 7$

$x = 1 - 2 y + z$

Multiply $- 2$ by $2$ .

$2 - 4 y + 2 z + 6 y + z = 8$

$1 + y + 3 z = 7$

$x = 1 - 2 y + z$

$2 - 4 y + 2 z + 6 y + z = 8$

$1 + y + 3 z = 7$

$x = 1 - 2 y + z$

$2 - 4 y + 2 z + 6 y + z = 8$

$1 + y + 3 z = 7$

$x = 1 - 2 y + z$

Simplify by adding terms.

Add $- 4 y$ and $6 y$ .

$2 + 2 y + 2 z + z = 8$

$1 + y + 3 z = 7$

$x = 1 - 2 y + z$

Add $2 z$ and $z$ .

$2 + 2 y + 3 z = 8$

$1 + y + 3 z = 7$

$x = 1 - 2 y + z$

$2 + 2 y + 3 z = 8$

$1 + y + 3 z = 7$

$x = 1 - 2 y + z$

$2 + 2 y + 3 z = 8$

$1 + y + 3 z = 7$

$x = 1 - 2 y + z$

$2 + 2 y + 3 z = 8$

$1 + y + 3 z = 7$

$x = 1 - 2 y + z$

$2 + 2 y + 3 z = 8$

$1 + y + 3 z = 7$

$x = 1 - 2 y + z$

Step 5

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

Subtract $1$ from both sides of the equation.

$y + 3 z = 7 - 1$

$2 + 2 y + 3 z = 8$

$x = 1 - 2 y + z$

Subtract $3 z$ from both sides of the equation.

$y = 7 - 1 - 3 z$

$2 + 2 y + 3 z = 8$

$x = 1 - 2 y + z$

Subtract $1$ from $7$ .

$y = 6 - 3 z$

$2 + 2 y + 3 z = 8$

$x = 1 - 2 y + z$

$y = 6 - 3 z$

$2 + 2 y + 3 z = 8$

$x = 1 - 2 y + z$

Step 6

Replace all occurrences of $y$ with $6 - 3 z$ in each equation.

Replace all occurrences of $y$ in $2 + 2 y + 3 z = 8$ with $6 - 3 z$ .

$2 + 2 (6 - 3 z) + 3 z = 8$

$y = 6 - 3 z$

$x = 1 - 2 y + z$

Simplify the left side.

Simplify $2 + 2 (6 - 3 z) + 3 z$ .

Simplify each term.

Apply the distributive property.

$2 + 2 \cdot 6 + 2 (- 3 z) + 3 z = 8$

$y = 6 - 3 z$

$x = 1 - 2 y + z$

Multiply $2$ by $6$ .

$2 + 12 + 2 (- 3 z) + 3 z = 8$

$y = 6 - 3 z$

$x = 1 - 2 y + z$

Multiply $- 3$ by $2$ .

$2 + 12 - 6 z + 3 z = 8$

$y = 6 - 3 z$

$x = 1 - 2 y + z$

$2 + 12 - 6 z + 3 z = 8$

$y = 6 - 3 z$

$x = 1 - 2 y + z$

Simplify by adding terms.

Add $2$ and $12$ .

$14 - 6 z + 3 z = 8$

$y = 6 - 3 z$

$x = 1 - 2 y + z$

Add $- 6 z$ and $3 z$ .

$14 - 3 z = 8$

$y = 6 - 3 z$

$x = 1 - 2 y + z$

$14 - 3 z = 8$

$y = 6 - 3 z$

$x = 1 - 2 y + z$

$14 - 3 z = 8$

$y = 6 - 3 z$

$x = 1 - 2 y + z$

$14 - 3 z = 8$

$y = 6 - 3 z$

$x = 1 - 2 y + z$

Replace all occurrences of $y$ in $x = 1 - 2 y + z$ with $6 - 3 z$ .

$x = 1 - 2 (6 - 3 z) + z$

$14 - 3 z = 8$

$y = 6 - 3 z$

Simplify the right side.

Simplify $1 - 2 (6 - 3 z) + z$ .

Simplify each term.

Apply the distributive property.

$x = 1 - 2 \cdot 6 - 2 (- 3 z) + z$

$14 - 3 z = 8$

$y = 6 - 3 z$

Multiply $- 2$ by $6$ .

$x = 1 - 12 - 2 (- 3 z) + z$

$14 - 3 z = 8$

$y = 6 - 3 z$

Multiply $- 3$ by $- 2$ .

$x = 1 - 12 + 6 z + z$

$14 - 3 z = 8$

$y = 6 - 3 z$

$x = 1 - 12 + 6 z + z$

$14 - 3 z = 8$

$y = 6 - 3 z$

Simplify by adding terms.

Subtract $12$ from $1$ .

$x = - 11 + 6 z + z$

$14 - 3 z = 8$

$y = 6 - 3 z$

Add $6 z$ and $z$ .

$x = - 11 + 7 z$

$14 - 3 z = 8$

$y = 6 - 3 z$

$x = - 11 + 7 z$

$14 - 3 z = 8$

$y = 6 - 3 z$

$x = - 11 + 7 z$

$14 - 3 z = 8$

$y = 6 - 3 z$

$x = - 11 + 7 z$

$14 - 3 z = 8$

$y = 6 - 3 z$

$x = - 11 + 7 z$

$14 - 3 z = 8$

$y = 6 - 3 z$

Step 7

Solve for $z$ in $14 - 3 z = 8$ .

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

Subtract $14$ from both sides of the equation.

$- 3 z = 8 - 14$

$x = - 11 + 7 z$

$y = 6 - 3 z$

Subtract $14$ from $8$ .

$- 3 z = - 6$

$x = - 11 + 7 z$

$y = 6 - 3 z$

$- 3 z = - 6$

$x = - 11 + 7 z$

$y = 6 - 3 z$

Divide each term in $- 3 z = - 6$ by $- 3$ and simplify.

Divide each term in $- 3 z = - 6$ by $- 3$ .

$\frac{- 3 z}{- 3} = \frac{- 6}{- 3}$

$x = - 11 + 7 z$

$y = 6 - 3 z$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 z}{- 3} = \frac{- 6}{- 3}$

$x = - 11 + 7 z$

$y = 6 - 3 z$

Divide $z$ by $1$ .

$z = \frac{- 6}{- 3}$

$x = - 11 + 7 z$

$y = 6 - 3 z$

$z = \frac{- 6}{- 3}$

$x = - 11 + 7 z$

$y = 6 - 3 z$

$z = \frac{- 6}{- 3}$

$x = - 11 + 7 z$

$y = 6 - 3 z$

Simplify the right side.

Divide $- 6$ by $- 3$ .

$z = 2$

$x = - 11 + 7 z$

$y = 6 - 3 z$

$z = 2$

$x = - 11 + 7 z$

$y = 6 - 3 z$

$z = 2$

$x = - 11 + 7 z$

$y = 6 - 3 z$

$z = 2$

$x = - 11 + 7 z$

$y = 6 - 3 z$

Step 8

Replace all occurrences of $z$ with $2$ in each equation.

Replace all occurrences of $z$ in $x = - 11 + 7 z$ with $2$ .

$x = - 11 + 7 (2)$

$z = 2$

$y = 6 - 3 z$

Simplify the right side.

Simplify $- 11 + 7 (2)$ .

Multiply $7$ by $2$ .

$x = - 11 + 14$

$z = 2$

$y = 6 - 3 z$

Add $- 11$ and $14$ .

$x = 3$

$z = 2$

$y = 6 - 3 z$

$x = 3$

$z = 2$

$y = 6 - 3 z$

$x = 3$

$z = 2$

$y = 6 - 3 z$

Replace all occurrences of $z$ in $y = 6 - 3 z$ with $2$ .

$y = 6 - 3 \cdot 2$

$x = 3$

$z = 2$

Simplify the right side.

Simplify $6 - 3 \cdot 2$ .

Multiply $- 3$ by $2$ .

$y = 6 - 6$

$x = 3$

$z = 2$

Subtract $6$ from $6$ .

$y = 0$

$x = 3$

$z = 2$

$y = 0$

$x = 3$

$z = 2$

$y = 0$

$x = 3$

$z = 2$

$y = 0$

$x = 3$

$z = 2$

Step 9

List all of the solutions.

$y = 0, x = 3, z = 2$

Find the Variables [[1,2,-1],[1,3,2],[2,6,1]]*[[x],[y],[z]]=[[1],[7],[8]]

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