Solve the Matrix Equation [[2,20,-4],[10,-6,8],[2,0,3]][[x],[y],[z]]=[[40],[16],[7]]

$[\begin{array}{c} 2 & 20 & - 4 \\ 10 & - 6 & 8 \\ 2 & 0 & 3 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 40 \\ 16 \\ 7 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 2 & 20 & - 4 \\ 10 & - 6 & 8 \\ 2 & 0 & 3 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}]$ .

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

$[\begin{array}{c} 2 x + 20 y - 4 z \\ 10 x - 6 y + 8 z \\ 2 x + 0 y + 3 z \end{array}] = [\begin{array}{c} 40 \\ 16 \\ 7 \end{array}]$

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

$[\begin{array}{c} 2 x + 20 y - 4 z \\ 10 x - 6 y + 8 z \\ 2 x + 3 z \end{array}] = [\begin{array}{c} 40 \\ 16 \\ 7 \end{array}]$

Step 2

Write as a linear system of equations.

$2 x + 20 y - 4 z = 40 10 x - 6 y + 8 z = 16 2 x + 3 z = 7$

Step 3

Solve the system of equations.

Solve for $x$ in $2 x + 20 y - 4 z = 40$ .

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

Subtract $20 y$ from both sides of the equation.

$2 x - 4 z = 40 - 20 y$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

Add $4 z$ to both sides of the equation.

$2 x = 40 - 20 y + 4 z$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

$2 x = 40 - 20 y + 4 z$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

Divide each term in $2 x = 40 - 20 y + 4 z$ by $2$ and simplify.

Divide each term in $2 x = 40 - 20 y + 4 z$ by $2$ .

$\frac{2 x}{2} = \frac{40}{2} + \frac{- 20 y}{2} + \frac{4 z}{2}$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{40}{2} + \frac{- 20 y}{2} + \frac{4 z}{2}$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

Divide $x$ by $1$ .

$x = \frac{40}{2} + \frac{- 20 y}{2} + \frac{4 z}{2}$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

$x = \frac{40}{2} + \frac{- 20 y}{2} + \frac{4 z}{2}$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

$x = \frac{40}{2} + \frac{- 20 y}{2} + \frac{4 z}{2}$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

Simplify the right side.

Simplify each term.

Divide $40$ by $2$ .

$x = 20 + \frac{- 20 y}{2} + \frac{4 z}{2}$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

Cancel the common factor of $- 20$ and $2$ .

Factor $2$ out of $- 20 y$ .

$x = 20 + \frac{2 (- 10 y)}{2} + \frac{4 z}{2}$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

Cancel the common factors.

Factor $2$ out of $2$ .

$x = 20 + \frac{2 (- 10 y)}{2 (1)} + \frac{4 z}{2}$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

Cancel the common factor.

$x = 20 + \frac{2 (- 10 y)}{2 \cdot 1} + \frac{4 z}{2}$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

Rewrite the expression.

$x = 20 + \frac{- 10 y}{1} + \frac{4 z}{2}$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

Divide $- 10 y$ by $1$ .

$x = 20 - 10 y + \frac{4 z}{2}$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

$x = 20 - 10 y + \frac{4 z}{2}$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

$x = 20 - 10 y + \frac{4 z}{2}$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4 z$ .

$x = 20 - 10 y + \frac{2 (2 z)}{2}$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

Cancel the common factors.

Factor $2$ out of $2$ .

$x = 20 - 10 y + \frac{2 (2 z)}{2 (1)}$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

Cancel the common factor.

$x = 20 - 10 y + \frac{2 (2 z)}{2 \cdot 1}$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

Rewrite the expression.

$x = 20 - 10 y + \frac{2 z}{1}$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

Divide $2 z$ by $1$ .

$x = 20 - 10 y + 2 z$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

$x = 20 - 10 y + 2 z$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

$x = 20 - 10 y + 2 z$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

$x = 20 - 10 y + 2 z$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

$x = 20 - 10 y + 2 z$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

$x = 20 - 10 y + 2 z$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

$x = 20 - 10 y + 2 z$

$10 x - 6 y + 8 z = 16$

$2 x + 3 z = 7$

Replace all occurrences of $x$ with $20 - 10 y + 2 z$ in each equation.

Replace all occurrences of $x$ in $10 x - 6 y + 8 z = 16$ with $20 - 10 y + 2 z$ .

$10 (20 - 10 y + 2 z) - 6 y + 8 z = 16$

$x = 20 - 10 y + 2 z$

$2 x + 3 z = 7$

Simplify the left side.

Simplify $10 (20 - 10 y + 2 z) - 6 y + 8 z$ .

Simplify each term.

Apply the distributive property.

$10 \cdot 20 + 10 (- 10 y) + 10 (2 z) - 6 y + 8 z = 16$

$x = 20 - 10 y + 2 z$

$2 x + 3 z = 7$

Simplify.

Multiply $10$ by $20$ .

$200 + 10 (- 10 y) + 10 (2 z) - 6 y + 8 z = 16$

$x = 20 - 10 y + 2 z$

$2 x + 3 z = 7$

Multiply $- 10$ by $10$ .

$200 - 100 y + 10 (2 z) - 6 y + 8 z = 16$

$x = 20 - 10 y + 2 z$

$2 x + 3 z = 7$

Multiply $2$ by $10$ .

$200 - 100 y + 20 z - 6 y + 8 z = 16$

$x = 20 - 10 y + 2 z$

$2 x + 3 z = 7$

$200 - 100 y + 20 z - 6 y + 8 z = 16$

$x = 20 - 10 y + 2 z$

$2 x + 3 z = 7$

$200 - 100 y + 20 z - 6 y + 8 z = 16$

$x = 20 - 10 y + 2 z$

$2 x + 3 z = 7$

Simplify by adding terms.

Subtract $6 y$ from $- 100 y$ .

$200 - 106 y + 20 z + 8 z = 16$

$x = 20 - 10 y + 2 z$

$2 x + 3 z = 7$

Add $20 z$ and $8 z$ .

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

$2 x + 3 z = 7$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

$2 x + 3 z = 7$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

$2 x + 3 z = 7$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

$2 x + 3 z = 7$

Replace all occurrences of $x$ in $2 x + 3 z = 7$ with $20 - 10 y + 2 z$ .

$2 (20 - 10 y + 2 z) + 3 z = 7$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

Simplify the left side.

Simplify $2 (20 - 10 y + 2 z) + 3 z$ .

Simplify each term.

Apply the distributive property.

$2 \cdot 20 + 2 (- 10 y) + 2 (2 z) + 3 z = 7$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

Simplify.

Multiply $2$ by $20$ .

$40 + 2 (- 10 y) + 2 (2 z) + 3 z = 7$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

Multiply $- 10$ by $2$ .

$40 - 20 y + 2 (2 z) + 3 z = 7$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

Multiply $2$ by $2$ .

$40 - 20 y + 4 z + 3 z = 7$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

$40 - 20 y + 4 z + 3 z = 7$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

$40 - 20 y + 4 z + 3 z = 7$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

Add $4 z$ and $3 z$ .

$40 - 20 y + 7 z = 7$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

$40 - 20 y + 7 z = 7$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

$40 - 20 y + 7 z = 7$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

$40 - 20 y + 7 z = 7$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

Solve for $y$ in $40 - 20 y + 7 z = 7$ .

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

Subtract $40$ from both sides of the equation.

$- 20 y + 7 z = 7 - 40$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

Subtract $7 z$ from both sides of the equation.

$- 20 y = 7 - 40 - 7 z$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

Subtract $40$ from $7$ .

$- 20 y = - 33 - 7 z$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

$- 20 y = - 33 - 7 z$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

Divide each term in $- 20 y = - 33 - 7 z$ by $- 20$ and simplify.

Divide each term in $- 20 y = - 33 - 7 z$ by $- 20$ .

$\frac{- 20 y}{- 20} = \frac{- 33}{- 20} + \frac{- 7 z}{- 20}$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

Simplify the left side.

Cancel the common factor of $- 20$ .

Cancel the common factor.

$\frac{- 20 y}{- 20} = \frac{- 33}{- 20} + \frac{- 7 z}{- 20}$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

Divide $y$ by $1$ .

$y = \frac{- 33}{- 20} + \frac{- 7 z}{- 20}$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

$y = \frac{- 33}{- 20} + \frac{- 7 z}{- 20}$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

$y = \frac{- 33}{- 20} + \frac{- 7 z}{- 20}$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

Simplify the right side.

Simplify each term.

Dividing two negative values results in a positive value.

$y = \frac{33}{20} + \frac{- 7 z}{- 20}$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

Dividing two negative values results in a positive value.

$y = \frac{33}{20} + \frac{7 z}{20}$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

$y = \frac{33}{20} + \frac{7 z}{20}$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

$y = \frac{33}{20} + \frac{7 z}{20}$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

$y = \frac{33}{20} + \frac{7 z}{20}$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

$y = \frac{33}{20} + \frac{7 z}{20}$

$200 - 106 y + 28 z = 16$

$x = 20 - 10 y + 2 z$

Replace all occurrences of $y$ with $\frac{33}{20} + \frac{7 z}{20}$ in each equation.

Replace all occurrences of $y$ in $200 - 106 y + 28 z = 16$ with $\frac{33}{20} + \frac{7 z}{20}$ .

$200 - 106 (\frac{33}{20} + \frac{7 z}{20}) + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Simplify the left side.

Simplify $200 - 106 (\frac{33}{20} + \frac{7 z}{20}) + 28 z$ .

Simplify each term.

Apply the distributive property.

$200 - 106 (\frac{33}{20}) - 106 \frac{7 z}{20} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Cancel the common factor of $2$ .

Factor $2$ out of $- 106$ .

$200 + 2 (- 53) (\frac{33}{20}) - 106 \frac{7 z}{20} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Factor $2$ out of $20$ .

$200 + 2 \cdot (- 53 \frac{33}{2 \cdot 10}) - 106 \frac{7 z}{20} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Cancel the common factor.

$200 + 2 \cdot (- 53 \frac{33}{2 \cdot 10}) - 106 \frac{7 z}{20} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Rewrite the expression.

$200 - 53 (\frac{33}{10}) - 106 \frac{7 z}{20} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

$200 - 53 (\frac{33}{10}) - 106 \frac{7 z}{20} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Combine $- 53$ and $\frac{33}{10}$ .

$200 + \frac{- 53 \cdot 33}{10} - 106 \frac{7 z}{20} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Multiply $- 53$ by $33$ .

$200 + \frac{- 1749}{10} - 106 \frac{7 z}{20} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Cancel the common factor of $2$ .

Factor $2$ out of $- 106$ .

$200 + \frac{- 1749}{10} + 2 (- 53) (\frac{7 z}{20}) + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Factor $2$ out of $20$ .

$200 + \frac{- 1749}{10} + 2 \cdot (- 53 \frac{7 z}{2 \cdot 10}) + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Cancel the common factor.

$200 + \frac{- 1749}{10} + 2 \cdot (- 53 \frac{7 z}{2 \cdot 10}) + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Rewrite the expression.

$200 + \frac{- 1749}{10} - 53 \frac{7 z}{10} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

$200 + \frac{- 1749}{10} - 53 \frac{7 z}{10} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Combine $- 53$ and $\frac{7 z}{10}$ .

$200 + \frac{- 1749}{10} + \frac{- 53 (7 z)}{10} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Multiply $7$ by $- 53$ .

$200 + \frac{- 1749}{10} + \frac{- 371 z}{10} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Simplify each term.

Move the negative in front of the fraction.

$200 - \frac{1749}{10} + \frac{- 371 z}{10} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Move the negative in front of the fraction.

$200 - \frac{1749}{10} - \frac{371 z}{10} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

$200 - \frac{1749}{10} - \frac{371 z}{10} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

$200 - \frac{1749}{10} - \frac{371 z}{10} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

To write $200$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$200 \cdot \frac{10}{10} - \frac{1749}{10} - \frac{371 z}{10} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Combine $200$ and $\frac{10}{10}$ .

$\frac{200 \cdot 10}{10} - \frac{1749}{10} - \frac{371 z}{10} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Combine the numerators over the common denominator.

$\frac{200 \cdot 10 - 1749}{10} - \frac{371 z}{10} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Simplify the numerator.

Multiply $200$ by $10$ .

$\frac{2000 - 1749}{10} - \frac{371 z}{10} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Subtract $1749$ from $2000$ .

$\frac{251}{10} - \frac{371 z}{10} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

$\frac{251}{10} - \frac{371 z}{10} + 28 z = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

To write $28 z$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{251}{10} - \frac{371 z}{10} + 28 z \cdot \frac{10}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Combine $28 z$ and $\frac{10}{10}$ .

$\frac{251}{10} - \frac{371 z}{10} + \frac{28 z \cdot 10}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Combine the numerators over the common denominator.

$\frac{251}{10} + \frac{- 371 z + 28 z \cdot 10}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Combine the numerators over the common denominator.

$\frac{251 - 371 z + 28 z \cdot 10}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Multiply $10$ by $28$ .

$\frac{251 - 371 z + 280 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Add $- 371 z$ and $280 z$ .

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 10 y + 2 z$

Replace all occurrences of $y$ in $x = 20 - 10 y + 2 z$ with $\frac{33}{20} + \frac{7 z}{20}$ .

$x = 20 - 10 (\frac{33}{20} + \frac{7 z}{20}) + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Simplify the right side.

Simplify $20 - 10 (\frac{33}{20} + \frac{7 z}{20}) + 2 z$ .

Simplify each term.

Apply the distributive property.

$x = 20 - 10 (\frac{33}{20}) - 10 \frac{7 z}{20} + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Cancel the common factor of $10$ .

Factor $10$ out of $- 10$ .

$x = 20 + 10 (- 1) (\frac{33}{20}) - 10 \frac{7 z}{20} + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Factor $10$ out of $20$ .

$x = 20 + 10 \cdot (- 1 \frac{33}{10 \cdot 2}) - 10 \frac{7 z}{20} + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Cancel the common factor.

$x = 20 + 10 \cdot (- 1 \frac{33}{10 \cdot 2}) - 10 \frac{7 z}{20} + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Rewrite the expression.

$x = 20 - 1 (\frac{33}{2}) - 10 \frac{7 z}{20} + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 1 (\frac{33}{2}) - 10 \frac{7 z}{20} + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Cancel the common factor of $10$ .

Factor $10$ out of $- 10$ .

$x = 20 - 1 (\frac{33}{2}) + 10 (- 1) (\frac{7 z}{20}) + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Factor $10$ out of $20$ .

$x = 20 - 1 (\frac{33}{2}) + 10 \cdot (- 1 \frac{7 z}{10 \cdot 2}) + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Cancel the common factor.

$x = 20 - 1 (\frac{33}{2}) + 10 \cdot (- 1 \frac{7 z}{10 \cdot 2}) + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Rewrite the expression.

$x = 20 - 1 (\frac{33}{2}) - 1 \frac{7 z}{2} + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - 1 (\frac{33}{2}) - 1 \frac{7 z}{2} + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Simplify each term.

Rewrite $- 1 (\frac{33}{2})$ as $- (\frac{33}{2})$ .

$x = 20 - (\frac{33}{2}) - 1 \frac{7 z}{2} + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Rewrite $- 1 \frac{7 z}{2}$ as $- \frac{7 z}{2}$ .

$x = 20 - \frac{33}{2} - \frac{7 z}{2} + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - \frac{33}{2} - \frac{7 z}{2} + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 20 - \frac{33}{2} - \frac{7 z}{2} + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

To write $20$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = 20 \cdot \frac{2}{2} - \frac{33}{2} - \frac{7 z}{2} + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Combine $20$ and $\frac{2}{2}$ .

$x = \frac{20 \cdot 2}{2} - \frac{33}{2} - \frac{7 z}{2} + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Combine the numerators over the common denominator.

$x = \frac{20 \cdot 2 - 33}{2} - \frac{7 z}{2} + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Simplify the numerator.

Multiply $20$ by $2$ .

$x = \frac{40 - 33}{2} - \frac{7 z}{2} + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Subtract $33$ from $40$ .

$x = \frac{7}{2} - \frac{7 z}{2} + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = \frac{7}{2} - \frac{7 z}{2} + 2 z$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

To write $2 z$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{7}{2} - \frac{7 z}{2} + 2 z \cdot \frac{2}{2}$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Combine $2 z$ and $\frac{2}{2}$ .

$x = \frac{7}{2} - \frac{7 z}{2} + \frac{2 z \cdot 2}{2}$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Combine the numerators over the common denominator.

$x = \frac{7}{2} + \frac{- 7 z + 2 z \cdot 2}{2}$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Combine the numerators over the common denominator.

$x = \frac{7 - 7 z + 2 z \cdot 2}{2}$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Multiply $2$ by $2$ .

$x = \frac{7 - 7 z + 4 z}{2}$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Add $- 7 z$ and $4 z$ .

$x = \frac{7 - 3 z}{2}$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = \frac{7 - 3 z}{2}$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = \frac{7 - 3 z}{2}$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = \frac{7 - 3 z}{2}$

$\frac{251 - 91 z}{10} = 16$

$y = \frac{33}{20} + \frac{7 z}{20}$

Solve for $z$ in $\frac{251 - 91 z}{10} = 16$ .

Multiply both sides by $10$ .

$\frac{251 - 91 z}{10} \cdot 10 = 16 \cdot 10$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

Simplify.

Simplify the left side.

Simplify $\frac{251 - 91 z}{10} \cdot 10$ .

Cancel the common factor of $10$ .

Cancel the common factor.

$\frac{251 - 91 z}{10} \cdot 10 = 16 \cdot 10$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

Rewrite the expression.

$251 - 91 z = 16 \cdot 10$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

$251 - 91 z = 16 \cdot 10$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

Reorder $251$ and $- 91 z$ .

$- 91 z + 251 = 16 \cdot 10$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

$- 91 z + 251 = 16 \cdot 10$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

$- 91 z + 251 = 16 \cdot 10$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

Simplify the right side.

Multiply $16$ by $10$ .

$- 91 z + 251 = 160$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

$- 91 z + 251 = 160$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

$- 91 z + 251 = 160$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

Solve for $z$ .

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

Subtract $251$ from both sides of the equation.

$- 91 z = 160 - 251$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

Subtract $251$ from $160$ .

$- 91 z = - 91$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

$- 91 z = - 91$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

Divide each term in $- 91 z = - 91$ by $- 91$ and simplify.

Divide each term in $- 91 z = - 91$ by $- 91$ .

$\frac{- 91 z}{- 91} = \frac{- 91}{- 91}$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

Simplify the left side.

Cancel the common factor of $- 91$ .

Cancel the common factor.

$\frac{- 91 z}{- 91} = \frac{- 91}{- 91}$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

Divide $z$ by $1$ .

$z = \frac{- 91}{- 91}$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

$z = \frac{- 91}{- 91}$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

$z = \frac{- 91}{- 91}$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

Simplify the right side.

Divide $- 91$ by $- 91$ .

$z = 1$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

$z = 1$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

$z = 1$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

$z = 1$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

$z = 1$

$x = \frac{7 - 3 z}{2}$

$y = \frac{33}{20} + \frac{7 z}{20}$

Replace all occurrences of $z$ with $1$ in each equation.

Replace all occurrences of $z$ in $x = \frac{7 - 3 z}{2}$ with $1$ .

$x = \frac{7 - 3 \cdot 1}{2}$

$z = 1$

$y = \frac{33}{20} + \frac{7 z}{20}$

Simplify the right side.

Simplify $\frac{7 - 3 \cdot 1}{2}$ .

Simplify the numerator.

Multiply $- 3$ by $1$ .

$x = \frac{7 - 3}{2}$

$z = 1$

$y = \frac{33}{20} + \frac{7 z}{20}$

Subtract $3$ from $7$ .

$x = \frac{4}{2}$

$z = 1$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = \frac{4}{2}$

$z = 1$

$y = \frac{33}{20} + \frac{7 z}{20}$

Divide $4$ by $2$ .

$x = 2$

$z = 1$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 2$

$z = 1$

$y = \frac{33}{20} + \frac{7 z}{20}$

$x = 2$

$z = 1$

$y = \frac{33}{20} + \frac{7 z}{20}$

Replace all occurrences of $z$ in $y = \frac{33}{20} + \frac{7 z}{20}$ with $1$ .

$y = \frac{33}{20} + \frac{7 (1)}{20}$

$x = 2$

$z = 1$

Simplify the right side.

Simplify $\frac{33}{20} + \frac{7 (1)}{20}$ .

Combine the numerators over the common denominator.

$y = \frac{33 + 7 (1)}{20}$

$x = 2$

$z = 1$

Simplify the expression.

Multiply $7$ by $1$ .

$y = \frac{33 + 7}{20}$

$x = 2$

$z = 1$

Add $33$ and $7$ .

$y = \frac{40}{20}$

$x = 2$

$z = 1$

Divide $40$ by $20$ .

$y = 2$

$x = 2$

$z = 1$

$y = 2$

$x = 2$

$z = 1$

$y = 2$

$x = 2$

$z = 1$

$y = 2$

$x = 2$

$z = 1$

$y = 2$

$x = 2$

$z = 1$

List all of the solutions.

$y = 2, x = 2, z = 1$

Solve the Matrix Equation [[2,20,-4],[10,-6,8],[2,0,3]][[x],[y],[z]]=[[40],[16],[7]]

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