Solve the System of Equations 2x+2y+5z=-1 2x-y+z=2 2x+4y-3z=14

$2 x + 2 y + 5 z = - 1$ $2 x - y + z = 2$ $2 x + 4 y - 3 z = 14$

Step 1

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

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

$- y + z = 2 - 2 x$

$2 x + 2 y + 5 z = - 1$

$2 x + 4 y - 3 z = 14$

Add $y$ to both sides of the equation.

$z = 2 - 2 x + y$

$2 x + 2 y + 5 z = - 1$

$2 x + 4 y - 3 z = 14$

$z = 2 - 2 x + y$

$2 x + 2 y + 5 z = - 1$

$2 x + 4 y - 3 z = 14$

Step 2

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

Replace all occurrences of $z$ in $2 x + 2 y + 5 z = - 1$ with $2 - 2 x + y$ .

$2 x + 2 y + 5 (2 - 2 x + y) = - 1$

$z = 2 - 2 x + y$

$2 x + 4 y - 3 z = 14$

Simplify the left side.

Simplify $2 x + 2 y + 5 (2 - 2 x + y)$ .

Simplify each term.

Apply the distributive property.

$2 x + 2 y + 5 \cdot 2 + 5 (- 2 x) + 5 y = - 1$

$z = 2 - 2 x + y$

$2 x + 4 y - 3 z = 14$

Simplify.

Multiply $5$ by $2$ .

$2 x + 2 y + 10 + 5 (- 2 x) + 5 y = - 1$

$z = 2 - 2 x + y$

$2 x + 4 y - 3 z = 14$

Multiply $- 2$ by $5$ .

$2 x + 2 y + 10 - 10 x + 5 y = - 1$

$z = 2 - 2 x + y$

$2 x + 4 y - 3 z = 14$

$2 x + 2 y + 10 - 10 x + 5 y = - 1$

$z = 2 - 2 x + y$

$2 x + 4 y - 3 z = 14$

$2 x + 2 y + 10 - 10 x + 5 y = - 1$

$z = 2 - 2 x + y$

$2 x + 4 y - 3 z = 14$

Simplify by adding terms.

Subtract $10 x$ from $2 x$ .

$- 8 x + 2 y + 10 + 5 y = - 1$

$z = 2 - 2 x + y$

$2 x + 4 y - 3 z = 14$

Add $2 y$ and $5 y$ .

$- 8 x + 7 y + 10 = - 1$

$z = 2 - 2 x + y$

$2 x + 4 y - 3 z = 14$

$- 8 x + 7 y + 10 = - 1$

$z = 2 - 2 x + y$

$2 x + 4 y - 3 z = 14$

$- 8 x + 7 y + 10 = - 1$

$z = 2 - 2 x + y$

$2 x + 4 y - 3 z = 14$

$- 8 x + 7 y + 10 = - 1$

$z = 2 - 2 x + y$

$2 x + 4 y - 3 z = 14$

Replace all occurrences of $z$ in $2 x + 4 y - 3 z = 14$ with $2 - 2 x + y$ .

$2 x + 4 y - 3 (2 - 2 x + y) = 14$

$- 8 x + 7 y + 10 = - 1$

$z = 2 - 2 x + y$

Simplify the left side.

Simplify $2 x + 4 y - 3 (2 - 2 x + y)$ .

Simplify each term.

Apply the distributive property.

$2 x + 4 y - 3 \cdot 2 - 3 (- 2 x) - 3 y = 14$

$- 8 x + 7 y + 10 = - 1$

$z = 2 - 2 x + y$

Simplify.

Multiply $- 3$ by $2$ .

$2 x + 4 y - 6 - 3 (- 2 x) - 3 y = 14$

$- 8 x + 7 y + 10 = - 1$

$z = 2 - 2 x + y$

Multiply $- 2$ by $- 3$ .

$2 x + 4 y - 6 + 6 x - 3 y = 14$

$- 8 x + 7 y + 10 = - 1$

$z = 2 - 2 x + y$

$2 x + 4 y - 6 + 6 x - 3 y = 14$

$- 8 x + 7 y + 10 = - 1$

$z = 2 - 2 x + y$

$2 x + 4 y - 6 + 6 x - 3 y = 14$

$- 8 x + 7 y + 10 = - 1$

$z = 2 - 2 x + y$

Simplify by adding terms.

Add $2 x$ and $6 x$ .

$8 x + 4 y - 6 - 3 y = 14$

$- 8 x + 7 y + 10 = - 1$

$z = 2 - 2 x + y$

Subtract $3 y$ from $4 y$ .

$8 x + y - 6 = 14$

$- 8 x + 7 y + 10 = - 1$

$z = 2 - 2 x + y$

$8 x + y - 6 = 14$

$- 8 x + 7 y + 10 = - 1$

$z = 2 - 2 x + y$

$8 x + y - 6 = 14$

$- 8 x + 7 y + 10 = - 1$

$z = 2 - 2 x + y$

$8 x + y - 6 = 14$

$- 8 x + 7 y + 10 = - 1$

$z = 2 - 2 x + y$

$8 x + y - 6 = 14$

$- 8 x + 7 y + 10 = - 1$

$z = 2 - 2 x + y$

Step 3

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

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

$y - 6 = 14 - 8 x$

$- 8 x + 7 y + 10 = - 1$

$z = 2 - 2 x + y$

Add $6$ to both sides of the equation.

$y = 14 - 8 x + 6$

$- 8 x + 7 y + 10 = - 1$

$z = 2 - 2 x + y$

Add $14$ and $6$ .

$y = - 8 x + 20$

$- 8 x + 7 y + 10 = - 1$

$z = 2 - 2 x + y$

$y = - 8 x + 20$

$- 8 x + 7 y + 10 = - 1$

$z = 2 - 2 x + y$

Step 4

Replace all occurrences of $y$ with $- 8 x + 20$ in each equation.

Replace all occurrences of $y$ in $- 8 x + 7 y + 10 = - 1$ with $- 8 x + 20$ .

$- 8 x + 7 (- 8 x + 20) + 10 = - 1$

$y = - 8 x + 20$

$z = 2 - 2 x + y$

Simplify the left side.

Simplify $- 8 x + 7 (- 8 x + 20) + 10$ .

Simplify each term.

Apply the distributive property.

$- 8 x + 7 (- 8 x) + 7 \cdot 20 + 10 = - 1$

$y = - 8 x + 20$

$z = 2 - 2 x + y$

Multiply $- 8$ by $7$ .

$- 8 x - 56 x + 7 \cdot 20 + 10 = - 1$

$y = - 8 x + 20$

$z = 2 - 2 x + y$

Multiply $7$ by $20$ .

$- 8 x - 56 x + 140 + 10 = - 1$

$y = - 8 x + 20$

$z = 2 - 2 x + y$

$- 8 x - 56 x + 140 + 10 = - 1$

$y = - 8 x + 20$

$z = 2 - 2 x + y$

Simplify by adding terms.

Subtract $56 x$ from $- 8 x$ .

$- 64 x + 140 + 10 = - 1$

$y = - 8 x + 20$

$z = 2 - 2 x + y$

Add $140$ and $10$ .

$- 64 x + 150 = - 1$

$y = - 8 x + 20$

$z = 2 - 2 x + y$

$- 64 x + 150 = - 1$

$y = - 8 x + 20$

$z = 2 - 2 x + y$

$- 64 x + 150 = - 1$

$y = - 8 x + 20$

$z = 2 - 2 x + y$

$- 64 x + 150 = - 1$

$y = - 8 x + 20$

$z = 2 - 2 x + y$

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

$z = 2 - 2 x - 8 x + 20$

$- 64 x + 150 = - 1$

$y = - 8 x + 20$

Simplify $z = 2 - 2 x - 8 x + 20$ .

Simplify the left side.

Remove parentheses.

$z = 2 - 2 x - 8 x + 20$

$- 64 x + 150 = - 1$

$y = - 8 x + 20$

$z = 2 - 2 x - 8 x + 20$

$- 64 x + 150 = - 1$

$y = - 8 x + 20$

Simplify the right side.

Simplify $2 - 2 x - 8 x + 20$ .

Add $2$ and $20$ .

$z = - 2 x - 8 x + 22$

$- 64 x + 150 = - 1$

$y = - 8 x + 20$

Subtract $8 x$ from $- 2 x$ .

$z = - 10 x + 22$

$- 64 x + 150 = - 1$

$y = - 8 x + 20$

$z = - 10 x + 22$

$- 64 x + 150 = - 1$

$y = - 8 x + 20$

$z = - 10 x + 22$

$- 64 x + 150 = - 1$

$y = - 8 x + 20$

$z = - 10 x + 22$

$- 64 x + 150 = - 1$

$y = - 8 x + 20$

$z = - 10 x + 22$

$- 64 x + 150 = - 1$

$y = - 8 x + 20$

Step 5

Solve for $x$ in $- 64 x + 150 = - 1$ .

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

Subtract $150$ from both sides of the equation.

$- 64 x = - 1 - 150$

$z = - 10 x + 22$

$y = - 8 x + 20$

Subtract $150$ from $- 1$ .

$- 64 x = - 151$

$z = - 10 x + 22$

$y = - 8 x + 20$

$- 64 x = - 151$

$z = - 10 x + 22$

$y = - 8 x + 20$

Divide each term in $- 64 x = - 151$ by $- 64$ and simplify.

Divide each term in $- 64 x = - 151$ by $- 64$ .

$\frac{- 64 x}{- 64} = \frac{- 151}{- 64}$

$z = - 10 x + 22$

$y = - 8 x + 20$

Simplify the left side.

Cancel the common factor of $- 64$ .

Cancel the common factor.

$\frac{- 64 x}{- 64} = \frac{- 151}{- 64}$

$z = - 10 x + 22$

$y = - 8 x + 20$

Divide $x$ by $1$ .

$x = \frac{- 151}{- 64}$

$z = - 10 x + 22$

$y = - 8 x + 20$

$x = \frac{- 151}{- 64}$

$z = - 10 x + 22$

$y = - 8 x + 20$

$x = \frac{- 151}{- 64}$

$z = - 10 x + 22$

$y = - 8 x + 20$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{151}{64}$

$z = - 10 x + 22$

$y = - 8 x + 20$

$x = \frac{151}{64}$

$z = - 10 x + 22$

$y = - 8 x + 20$

$x = \frac{151}{64}$

$z = - 10 x + 22$

$y = - 8 x + 20$

$x = \frac{151}{64}$

$z = - 10 x + 22$

$y = - 8 x + 20$

Step 6

Replace all occurrences of $x$ with $\frac{151}{64}$ in each equation.

Replace all occurrences of $x$ in $z = - 10 x + 22$ with $\frac{151}{64}$ .

$z = - 10 (\frac{151}{64}) + 22$

$x = \frac{151}{64}$

$y = - 8 x + 20$

Simplify the right side.

Simplify $- 10 (\frac{151}{64}) + 22$ .

Simplify each term.

Cancel the common factor of $2$ .

Factor $2$ out of $- 10$ .

$z = 2 (- 5) (\frac{151}{64}) + 22$

$x = \frac{151}{64}$

$y = - 8 x + 20$

Factor $2$ out of $64$ .

$z = 2 \cdot (- 5 \frac{151}{2 \cdot 32}) + 22$

$x = \frac{151}{64}$

$y = - 8 x + 20$

Cancel the common factor.

$z = 2 \cdot (- 5 \frac{151}{2 \cdot 32}) + 22$

$x = \frac{151}{64}$

$y = - 8 x + 20$

Rewrite the expression.

$z = - 5 (\frac{151}{32}) + 22$

$x = \frac{151}{64}$

$y = - 8 x + 20$

$z = - 5 (\frac{151}{32}) + 22$

$x = \frac{151}{64}$

$y = - 8 x + 20$

Combine $- 5$ and $\frac{151}{32}$ .

$z = \frac{- 5 \cdot 151}{32} + 22$

$x = \frac{151}{64}$

$y = - 8 x + 20$

Multiply $- 5$ by $151$ .

$z = \frac{- 755}{32} + 22$

$x = \frac{151}{64}$

$y = - 8 x + 20$

Move the negative in front of the fraction.

$z = - \frac{755}{32} + 22$

$x = \frac{151}{64}$

$y = - 8 x + 20$

$z = - \frac{755}{32} + 22$

$x = \frac{151}{64}$

$y = - 8 x + 20$

To write $22$ as a fraction with a common denominator, multiply by $\frac{32}{32}$ .

$z = - \frac{755}{32} + 22 \cdot \frac{32}{32}$

$x = \frac{151}{64}$

$y = - 8 x + 20$

Combine $22$ and $\frac{32}{32}$ .

$z = - \frac{755}{32} + \frac{22 \cdot 32}{32}$

$x = \frac{151}{64}$

$y = - 8 x + 20$

Combine the numerators over the common denominator.

$z = \frac{- 755 + 22 \cdot 32}{32}$

$x = \frac{151}{64}$

$y = - 8 x + 20$

Simplify the numerator.

Multiply $22$ by $32$ .

$z = \frac{- 755 + 704}{32}$

$x = \frac{151}{64}$

$y = - 8 x + 20$

Add $- 755$ and $704$ .

$z = \frac{- 51}{32}$

$x = \frac{151}{64}$

$y = - 8 x + 20$

$z = \frac{- 51}{32}$

$x = \frac{151}{64}$

$y = - 8 x + 20$

Move the negative in front of the fraction.

$z = - \frac{51}{32}$

$x = \frac{151}{64}$

$y = - 8 x + 20$

$z = - \frac{51}{32}$

$x = \frac{151}{64}$

$y = - 8 x + 20$

$z = - \frac{51}{32}$

$x = \frac{151}{64}$

$y = - 8 x + 20$

Replace all occurrences of $x$ in $y = - 8 x + 20$ with $\frac{151}{64}$ .

$y = - 8 (\frac{151}{64}) + 20$

$z = - \frac{51}{32}$

$x = \frac{151}{64}$

Simplify the right side.

Simplify $- 8 (\frac{151}{64}) + 20$ .

Simplify each term.

Cancel the common factor of $8$ .

Factor $8$ out of $- 8$ .

$y = 8 (- 1) (\frac{151}{64}) + 20$

$z = - \frac{51}{32}$

$x = \frac{151}{64}$

Factor $8$ out of $64$ .

$y = 8 \cdot (- 1 \frac{151}{8 \cdot 8}) + 20$

$z = - \frac{51}{32}$

$x = \frac{151}{64}$

Cancel the common factor.

$y = 8 \cdot (- 1 \frac{151}{8 \cdot 8}) + 20$

$z = - \frac{51}{32}$

$x = \frac{151}{64}$

Rewrite the expression.

$y = - 1 (\frac{151}{8}) + 20$

$z = - \frac{51}{32}$

$x = \frac{151}{64}$

$y = - 1 (\frac{151}{8}) + 20$

$z = - \frac{51}{32}$

$x = \frac{151}{64}$

Rewrite $- 1 (\frac{151}{8})$ as $- (\frac{151}{8})$ .

$y = - \frac{151}{8} + 20$

$z = - \frac{51}{32}$

$x = \frac{151}{64}$

$y = - \frac{151}{8} + 20$

$z = - \frac{51}{32}$

$x = \frac{151}{64}$

To write $20$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = - \frac{151}{8} + 20 \cdot \frac{8}{8}$

$z = - \frac{51}{32}$

$x = \frac{151}{64}$

Combine $20$ and $\frac{8}{8}$ .

$y = - \frac{151}{8} + \frac{20 \cdot 8}{8}$

$z = - \frac{51}{32}$

$x = \frac{151}{64}$

Combine the numerators over the common denominator.

$y = \frac{- 151 + 20 \cdot 8}{8}$

$z = - \frac{51}{32}$

$x = \frac{151}{64}$

Simplify the numerator.

Multiply $20$ by $8$ .

$y = \frac{- 151 + 160}{8}$

$z = - \frac{51}{32}$

$x = \frac{151}{64}$

Add $- 151$ and $160$ .

$y = \frac{9}{8}$

$z = - \frac{51}{32}$

$x = \frac{151}{64}$

$y = \frac{9}{8}$

$z = - \frac{51}{32}$

$x = \frac{151}{64}$

$y = \frac{9}{8}$

$z = - \frac{51}{32}$

$x = \frac{151}{64}$

$y = \frac{9}{8}$

$z = - \frac{51}{32}$

$x = \frac{151}{64}$

$y = \frac{9}{8}$

$z = - \frac{51}{32}$

$x = \frac{151}{64}$

Step 7

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(\frac{151}{64}, \frac{9}{8}, - \frac{51}{32})$

Step 8

The result can be shown in multiple forms.

Point Form:

$(\frac{151}{64}, \frac{9}{8}, - \frac{51}{32})$

Equation Form:

$x = \frac{151}{64}, y = \frac{9}{8}, z = - \frac{51}{32}$

Solve the System of Equations 2x+2y+5z=-1 2x-y+z=2 2x+4y-3z=14

Related Questions

Support LCMGCF.com by disabling your adblocker.