Solve the System of Equations x-y+4z=10 3x+y+2z=4 2x+2y-3z=-7

$x - y + 4 z = 10$ $3 x + y + 2 z = 4$ $2 x + 2 y - 3 z = - 7$

Step 1

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

Add $y$ to both sides of the equation.

$x + 4 z = 10 + y$

$3 x + y + 2 z = 4$

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

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

$x = 10 + y - 4 z$

$3 x + y + 2 z = 4$

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

$x = 10 + y - 4 z$

$3 x + y + 2 z = 4$

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

Step 2

Replace all occurrences of $x$ with $10 + y - 4 z$ in each equation.

Replace all occurrences of $x$ in $3 x + y + 2 z = 4$ with $10 + y - 4 z$ .

$3 (10 + y - 4 z) + y + 2 z = 4$

$x = 10 + y - 4 z$

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

Simplify the left side.

Simplify $3 (10 + y - 4 z) + y + 2 z$ .

Simplify each term.

Apply the distributive property.

$3 \cdot 10 + 3 y + 3 (- 4 z) + y + 2 z = 4$

$x = 10 + y - 4 z$

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

Simplify.

Multiply $3$ by $10$ .

$30 + 3 y + 3 (- 4 z) + y + 2 z = 4$

$x = 10 + y - 4 z$

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

Multiply $- 4$ by $3$ .

$30 + 3 y - 12 z + y + 2 z = 4$

$x = 10 + y - 4 z$

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

$30 + 3 y - 12 z + y + 2 z = 4$

$x = 10 + y - 4 z$

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

$30 + 3 y - 12 z + y + 2 z = 4$

$x = 10 + y - 4 z$

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

Simplify by adding terms.

Add $3 y$ and $y$ .

$30 + 4 y - 12 z + 2 z = 4$

$x = 10 + y - 4 z$

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

Add $- 12 z$ and $2 z$ .

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

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

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

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

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

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

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

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

Replace all occurrences of $x$ in $2 x + 2 y - 3 z = - 7$ with $10 + y - 4 z$ .

$2 (10 + y - 4 z) + 2 y - 3 z = - 7$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

Simplify the left side.

Simplify $2 (10 + y - 4 z) + 2 y - 3 z$ .

Simplify each term.

Apply the distributive property.

$2 \cdot 10 + 2 y + 2 (- 4 z) + 2 y - 3 z = - 7$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

Simplify.

Multiply $2$ by $10$ .

$20 + 2 y + 2 (- 4 z) + 2 y - 3 z = - 7$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

Multiply $- 4$ by $2$ .

$20 + 2 y - 8 z + 2 y - 3 z = - 7$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

$20 + 2 y - 8 z + 2 y - 3 z = - 7$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

$20 + 2 y - 8 z + 2 y - 3 z = - 7$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

Simplify by adding terms.

Add $2 y$ and $2 y$ .

$20 + 4 y - 8 z - 3 z = - 7$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

Subtract $3 z$ from $- 8 z$ .

$20 + 4 y - 11 z = - 7$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

$20 + 4 y - 11 z = - 7$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

$20 + 4 y - 11 z = - 7$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

$20 + 4 y - 11 z = - 7$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

$20 + 4 y - 11 z = - 7$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

Step 3

Solve for $y$ in $20 + 4 y - 11 z = - 7$ .

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

Subtract $20$ from both sides of the equation.

$4 y - 11 z = - 7 - 20$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

Add $11 z$ to both sides of the equation.

$4 y = - 7 - 20 + 11 z$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

Subtract $20$ from $- 7$ .

$4 y = - 27 + 11 z$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

$4 y = - 27 + 11 z$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

Divide each term in $4 y = - 27 + 11 z$ by $4$ and simplify.

Divide each term in $4 y = - 27 + 11 z$ by $4$ .

$\frac{4 y}{4} = \frac{- 27}{4} + \frac{11 z}{4}$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{- 27}{4} + \frac{11 z}{4}$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

Divide $y$ by $1$ .

$y = \frac{- 27}{4} + \frac{11 z}{4}$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

$y = \frac{- 27}{4} + \frac{11 z}{4}$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

$y = \frac{- 27}{4} + \frac{11 z}{4}$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

Simplify the right side.

Move the negative in front of the fraction.

$y = - \frac{27}{4} + \frac{11 z}{4}$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$30 + 4 y - 10 z = 4$

$x = 10 + y - 4 z$

Step 4

Replace all occurrences of $y$ with $- \frac{27}{4} + \frac{11 z}{4}$ in each equation.

Replace all occurrences of $y$ in $30 + 4 y - 10 z = 4$ with $- \frac{27}{4} + \frac{11 z}{4}$ .

$30 + 4 (- \frac{27}{4} + \frac{11 z}{4}) - 10 z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = 10 + y - 4 z$

Simplify the left side.

Simplify $30 + 4 (- \frac{27}{4} + \frac{11 z}{4}) - 10 z$ .

Simplify each term.

Apply the distributive property.

$30 + 4 (- \frac{27}{4}) + 4 (\frac{11 z}{4}) - 10 z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = 10 + y - 4 z$

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{27}{4}$ into the numerator.

$30 + 4 (\frac{- 27}{4}) + 4 (\frac{11 z}{4}) - 10 z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = 10 + y - 4 z$

Cancel the common factor.

$30 + 4 (\frac{- 27}{4}) + 4 (\frac{11 z}{4}) - 10 z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = 10 + y - 4 z$

Rewrite the expression.

$30 - 27 + 4 (\frac{11 z}{4}) - 10 z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = 10 + y - 4 z$

$30 - 27 + 4 (\frac{11 z}{4}) - 10 z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = 10 + y - 4 z$

Cancel the common factor of $4$ .

Cancel the common factor.

$30 - 27 + 4 (\frac{11 z}{4}) - 10 z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = 10 + y - 4 z$

Rewrite the expression.

$30 - 27 + 11 z - 10 z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = 10 + y - 4 z$

$30 - 27 + 11 z - 10 z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = 10 + y - 4 z$

$30 - 27 + 11 z - 10 z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = 10 + y - 4 z$

Simplify by adding terms.

Subtract $27$ from $30$ .

$3 + 11 z - 10 z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = 10 + y - 4 z$

Subtract $10 z$ from $11 z$ .

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = 10 + y - 4 z$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = 10 + y - 4 z$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = 10 + y - 4 z$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = 10 + y - 4 z$

Replace all occurrences of $y$ in $x = 10 + y - 4 z$ with $- \frac{27}{4} + \frac{11 z}{4}$ .

$x = 10 - \frac{27}{4} + \frac{11 z}{4} - 4 z$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

Simplify $x = 10 - \frac{27}{4} + \frac{11 z}{4} - 4 z$ .

Simplify the left side.

Remove parentheses.

$x = 10 - \frac{27}{4} + \frac{11 z}{4} - 4 z$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = 10 - \frac{27}{4} + \frac{11 z}{4} - 4 z$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

Simplify the right side.

Simplify $10 - \frac{27}{4} + \frac{11 z}{4} - 4 z$ .

To write $10$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 10 \cdot \frac{4}{4} - \frac{27}{4} + \frac{11 z}{4} - 4 z$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

Combine $10$ and $\frac{4}{4}$ .

$x = \frac{10 \cdot 4}{4} - \frac{27}{4} + \frac{11 z}{4} - 4 z$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

Combine the numerators over the common denominator.

$x = \frac{10 \cdot 4 - 27}{4} + \frac{11 z}{4} - 4 z$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

Simplify the numerator.

Multiply $10$ by $4$ .

$x = \frac{40 - 27}{4} + \frac{11 z}{4} - 4 z$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

Subtract $27$ from $40$ .

$x = \frac{13}{4} + \frac{11 z}{4} - 4 z$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = \frac{13}{4} + \frac{11 z}{4} - 4 z$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

To write $- 4 z$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{13}{4} + \frac{11 z}{4} - 4 z \cdot \frac{4}{4}$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

Combine $- 4 z$ and $\frac{4}{4}$ .

$x = \frac{13}{4} + \frac{11 z}{4} + \frac{- 4 z \cdot 4}{4}$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

Combine the numerators over the common denominator.

$x = \frac{13}{4} + \frac{11 z - 4 z \cdot 4}{4}$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

Combine the numerators over the common denominator.

$x = \frac{13 + 11 z - 4 z \cdot 4}{4}$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

Multiply $4$ by $- 4$ .

$x = \frac{13 + 11 z - 16 z}{4}$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

Subtract $16 z$ from $11 z$ .

$x = \frac{13 - 5 z}{4}$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = \frac{13 - 5 z}{4}$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = \frac{13 - 5 z}{4}$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = \frac{13 - 5 z}{4}$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = \frac{13 - 5 z}{4}$

$3 + z = 4$

$y = - \frac{27}{4} + \frac{11 z}{4}$

Step 5

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

Subtract $3$ from both sides of the equation.

$z = 4 - 3$

$x = \frac{13 - 5 z}{4}$

$y = - \frac{27}{4} + \frac{11 z}{4}$

Subtract $3$ from $4$ .

$z = 1$

$x = \frac{13 - 5 z}{4}$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$z = 1$

$x = \frac{13 - 5 z}{4}$

$y = - \frac{27}{4} + \frac{11 z}{4}$

Step 6

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

Replace all occurrences of $z$ in $x = \frac{13 - 5 z}{4}$ with $1$ .

$x = \frac{13 - 5 \cdot 1}{4}$

$z = 1$

$y = - \frac{27}{4} + \frac{11 z}{4}$

Simplify the right side.

Simplify $\frac{13 - 5 \cdot 1}{4}$ .

Simplify the numerator.

Multiply $- 5$ by $1$ .

$x = \frac{13 - 5}{4}$

$z = 1$

$y = - \frac{27}{4} + \frac{11 z}{4}$

Subtract $5$ from $13$ .

$x = \frac{8}{4}$

$z = 1$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = \frac{8}{4}$

$z = 1$

$y = - \frac{27}{4} + \frac{11 z}{4}$

Divide $8$ by $4$ .

$x = 2$

$z = 1$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = 2$

$z = 1$

$y = - \frac{27}{4} + \frac{11 z}{4}$

$x = 2$

$z = 1$

$y = - \frac{27}{4} + \frac{11 z}{4}$

Replace all occurrences of $z$ in $y = - \frac{27}{4} + \frac{11 z}{4}$ with $1$ .

$y = - \frac{27}{4} + \frac{11 (1)}{4}$

$x = 2$

$z = 1$

Simplify the right side.

Simplify $- \frac{27}{4} + \frac{11 (1)}{4}$ .

Combine the numerators over the common denominator.

$y = \frac{- 27 + 11 (1)}{4}$

$x = 2$

$z = 1$

Simplify the expression.

Multiply $11$ by $1$ .

$y = \frac{- 27 + 11}{4}$

$x = 2$

$z = 1$

Add $- 27$ and $11$ .

$y = \frac{- 16}{4}$

$x = 2$

$z = 1$

Divide $- 16$ by $4$ .

$y = - 4$

$x = 2$

$z = 1$

$y = - 4$

$x = 2$

$z = 1$

$y = - 4$

$x = 2$

$z = 1$

$y = - 4$

$x = 2$

$z = 1$

$y = - 4$

$x = 2$

$z = 1$

Step 7

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

$(2, - 4, 1)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(2, - 4, 1)$

Equation Form:

$x = 2, y = - 4, z = 1$

Solve the System of Equations x-y+4z=10 3x+y+2z=4 2x+2y-3z=-7

Related Questions

Support LCMGCF.com by disabling your adblocker.