Solve the Matrix Equation [[1,4,2],[0,6,3],[2,7,1]][[x],[y],[z]]=[[2],[-3],[-3]]

$[\begin{array}{c} 1 & 4 & 2 \\ 0 & 6 & 3 \\ 2 & 7 & 1 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 2 \\ - 3 \\ - 3 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 1 & 4 & 2 \\ 0 & 6 & 3 \\ 2 & 7 & 1 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}]$ .

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

$[\begin{array}{c} 1 x + 4 y + 2 z \\ 0 x + 6 y + 3 z \\ 2 x + 7 y + 1 z \end{array}] = [\begin{array}{c} 2 \\ - 3 \\ - 3 \end{array}]$

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

$[\begin{array}{c} x + 4 y + 2 z \\ 6 y + 3 z \\ 2 x + 7 y + z \end{array}] = [\begin{array}{c} 2 \\ - 3 \\ - 3 \end{array}]$

Step 2

Write as a linear system of equations.

$x + 4 y + 2 z = 2 6 y + 3 z = - 3 2 x + 7 y + z = - 3$

Step 3

Solve the system of equations.

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

Subtract $4 y$ from both sides of the equation.

$x + 2 z = 2 - 4 y$

$6 y + 3 z = - 3$

$2 x + 7 y + z = - 3$

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

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

$2 x + 7 y + z = - 3$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

$2 x + 7 y + z = - 3$

Replace all occurrences of $x$ with $2 - 4 y - 2 z$ in each equation.

Replace all occurrences of $x$ in $2 x + 7 y + z = - 3$ with $2 - 4 y - 2 z$ .

$2 (2 - 4 y - 2 z) + 7 y + z = - 3$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

Simplify the left side.

Simplify $2 (2 - 4 y - 2 z) + 7 y + z$ .

Simplify each term.

Apply the distributive property.

$2 \cdot 2 + 2 (- 4 y) + 2 (- 2 z) + 7 y + z = - 3$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

Simplify.

Multiply $2$ by $2$ .

$4 + 2 (- 4 y) + 2 (- 2 z) + 7 y + z = - 3$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

Multiply $- 4$ by $2$ .

$4 - 8 y + 2 (- 2 z) + 7 y + z = - 3$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

Multiply $- 2$ by $2$ .

$4 - 8 y - 4 z + 7 y + z = - 3$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

$4 - 8 y - 4 z + 7 y + z = - 3$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

$4 - 8 y - 4 z + 7 y + z = - 3$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

Simplify by adding terms.

Add $- 8 y$ and $7 y$ .

$4 - y - 4 z + z = - 3$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

Add $- 4 z$ and $z$ .

$4 - y - 3 z = - 3$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

$4 - y - 3 z = - 3$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

$4 - y - 3 z = - 3$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

$4 - y - 3 z = - 3$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

$4 - y - 3 z = - 3$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

Solve for $y$ in $4 - y - 3 z = - 3$ .

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

Subtract $4$ from both sides of the equation.

$- y - 3 z = - 3 - 4$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

Add $3 z$ to both sides of the equation.

$- y = - 3 - 4 + 3 z$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

Subtract $4$ from $- 3$ .

$- y = - 7 + 3 z$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

$- y = - 7 + 3 z$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

Divide each term in $- y = - 7 + 3 z$ by $- 1$ and simplify.

Divide each term in $- y = - 7 + 3 z$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 7}{- 1} + \frac{3 z}{- 1}$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 7}{- 1} + \frac{3 z}{- 1}$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

Divide $y$ by $1$ .

$y = \frac{- 7}{- 1} + \frac{3 z}{- 1}$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

$y = \frac{- 7}{- 1} + \frac{3 z}{- 1}$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

Simplify the right side.

Simplify each term.

Divide $- 7$ by $- 1$ .

$y = 7 + \frac{3 z}{- 1}$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

Move the negative one from the denominator of $\frac{3 z}{- 1}$ .

$y = 7 - 1 \cdot (3 z)$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

Rewrite $- 1 \cdot (3 z)$ as $- (3 z)$ .

$y = 7 - (3 z)$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

Multiply $3$ by $- 1$ .

$y = 7 - 3 z$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

$y = 7 - 3 z$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

$y = 7 - 3 z$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

$y = 7 - 3 z$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

$y = 7 - 3 z$

$x = 2 - 4 y - 2 z$

$6 y + 3 z = - 3$

Replace all occurrences of $y$ with $7 - 3 z$ in each equation.

Replace all occurrences of $y$ in $x = 2 - 4 y - 2 z$ with $7 - 3 z$ .

$x = 2 - 4 (7 - 3 z) - 2 z$

$y = 7 - 3 z$

$6 y + 3 z = - 3$

Simplify the right side.

Simplify $2 - 4 (7 - 3 z) - 2 z$ .

Simplify each term.

Apply the distributive property.

$x = 2 - 4 \cdot 7 - 4 (- 3 z) - 2 z$

$y = 7 - 3 z$

$6 y + 3 z = - 3$

Multiply $- 4$ by $7$ .

$x = 2 - 28 - 4 (- 3 z) - 2 z$

$y = 7 - 3 z$

$6 y + 3 z = - 3$

Multiply $- 3$ by $- 4$ .

$x = 2 - 28 + 12 z - 2 z$

$y = 7 - 3 z$

$6 y + 3 z = - 3$

$x = 2 - 28 + 12 z - 2 z$

$y = 7 - 3 z$

$6 y + 3 z = - 3$

Simplify by adding terms.

Subtract $28$ from $2$ .

$x = - 26 + 12 z - 2 z$

$y = 7 - 3 z$

$6 y + 3 z = - 3$

Subtract $2 z$ from $12 z$ .

$x = - 26 + 10 z$

$y = 7 - 3 z$

$6 y + 3 z = - 3$

$x = - 26 + 10 z$

$y = 7 - 3 z$

$6 y + 3 z = - 3$

$x = - 26 + 10 z$

$y = 7 - 3 z$

$6 y + 3 z = - 3$

$x = - 26 + 10 z$

$y = 7 - 3 z$

$6 y + 3 z = - 3$

Replace all occurrences of $y$ in $6 y + 3 z = - 3$ with $7 - 3 z$ .

$6 (7 - 3 z) + 3 z = - 3$

$x = - 26 + 10 z$

$y = 7 - 3 z$

Simplify the left side.

Simplify $6 (7 - 3 z) + 3 z$ .

Simplify each term.

Apply the distributive property.

$6 \cdot 7 + 6 (- 3 z) + 3 z = - 3$

$x = - 26 + 10 z$

$y = 7 - 3 z$

Multiply $6$ by $7$ .

$42 + 6 (- 3 z) + 3 z = - 3$

$x = - 26 + 10 z$

$y = 7 - 3 z$

Multiply $- 3$ by $6$ .

$42 - 18 z + 3 z = - 3$

$x = - 26 + 10 z$

$y = 7 - 3 z$

$42 - 18 z + 3 z = - 3$

$x = - 26 + 10 z$

$y = 7 - 3 z$

Add $- 18 z$ and $3 z$ .

$42 - 15 z = - 3$

$x = - 26 + 10 z$

$y = 7 - 3 z$

$42 - 15 z = - 3$

$x = - 26 + 10 z$

$y = 7 - 3 z$

$42 - 15 z = - 3$

$x = - 26 + 10 z$

$y = 7 - 3 z$

$42 - 15 z = - 3$

$x = - 26 + 10 z$

$y = 7 - 3 z$

Solve for $z$ in $42 - 15 z = - 3$ .

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

Subtract $42$ from both sides of the equation.

$- 15 z = - 3 - 42$

$x = - 26 + 10 z$

$y = 7 - 3 z$

Subtract $42$ from $- 3$ .

$- 15 z = - 45$

$x = - 26 + 10 z$

$y = 7 - 3 z$

$- 15 z = - 45$

$x = - 26 + 10 z$

$y = 7 - 3 z$

Divide each term in $- 15 z = - 45$ by $- 15$ and simplify.

Divide each term in $- 15 z = - 45$ by $- 15$ .

$\frac{- 15 z}{- 15} = \frac{- 45}{- 15}$

$x = - 26 + 10 z$

$y = 7 - 3 z$

Simplify the left side.

Cancel the common factor of $- 15$ .

Cancel the common factor.

$\frac{- 15 z}{- 15} = \frac{- 45}{- 15}$

$x = - 26 + 10 z$

$y = 7 - 3 z$

Divide $z$ by $1$ .

$z = \frac{- 45}{- 15}$

$x = - 26 + 10 z$

$y = 7 - 3 z$

$z = \frac{- 45}{- 15}$

$x = - 26 + 10 z$

$y = 7 - 3 z$

$z = \frac{- 45}{- 15}$

$x = - 26 + 10 z$

$y = 7 - 3 z$

Simplify the right side.

Divide $- 45$ by $- 15$ .

$z = 3$

$x = - 26 + 10 z$

$y = 7 - 3 z$

$z = 3$

$x = - 26 + 10 z$

$y = 7 - 3 z$

$z = 3$

$x = - 26 + 10 z$

$y = 7 - 3 z$

$z = 3$

$x = - 26 + 10 z$

$y = 7 - 3 z$

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

Replace all occurrences of $z$ in $x = - 26 + 10 z$ with $3$ .

$x = - 26 + 10 (3)$

$z = 3$

$y = 7 - 3 z$

Simplify the right side.

Simplify $- 26 + 10 (3)$ .

Multiply $10$ by $3$ .

$x = - 26 + 30$

$z = 3$

$y = 7 - 3 z$

Add $- 26$ and $30$ .

$x = 4$

$z = 3$

$y = 7 - 3 z$

$x = 4$

$z = 3$

$y = 7 - 3 z$

$x = 4$

$z = 3$

$y = 7 - 3 z$

Replace all occurrences of $z$ in $y = 7 - 3 z$ with $3$ .

$y = 7 - 3 \cdot 3$

$x = 4$

$z = 3$

Simplify the right side.

Simplify $7 - 3 \cdot 3$ .

Multiply $- 3$ by $3$ .

$y = 7 - 9$

$x = 4$

$z = 3$

Subtract $9$ from $7$ .

$y = - 2$

$x = 4$

$z = 3$

$y = - 2$

$x = 4$

$z = 3$

$y = - 2$

$x = 4$

$z = 3$

$y = - 2$

$x = 4$

$z = 3$

List all of the solutions.

$y = - 2, x = 4, z = 3$

Solve the Matrix Equation [[1,4,2],[0,6,3],[2,7,1]][[x],[y],[z]]=[[2],[-3],[-3]]

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