Solve the System of Equations 2x-3y+z=4 -x+5y-z=-1 5x-4y+2z=11

$2 x - 3 y + z = 4$ $- x + 5 y - z = - 1$ $5 x - 4 y + 2 z = 11$

Step 1

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

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

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

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

$5 x - 4 y + 2 z = 11$

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

$z = 4 - 2 x + 3 y$

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

$5 x - 4 y + 2 z = 11$

$z = 4 - 2 x + 3 y$

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

$5 x - 4 y + 2 z = 11$

Step 2

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

Replace all occurrences of $z$ in $- x + 5 y - z = - 1$ with $4 - 2 x + 3 y$ .

$- x + 5 y - (4 - 2 x + 3 y) = - 1$

$z = 4 - 2 x + 3 y$

$5 x - 4 y + 2 z = 11$

Simplify the left side.

Simplify $- x + 5 y - (4 - 2 x + 3 y)$ .

Simplify each term.

Apply the distributive property.

$- x + 5 y - 1 \cdot 4 - (- 2 x) - (3 y) = - 1$

$z = 4 - 2 x + 3 y$

$5 x - 4 y + 2 z = 11$

Simplify.

Multiply $- 1$ by $4$ .

$- x + 5 y - 4 - (- 2 x) - (3 y) = - 1$

$z = 4 - 2 x + 3 y$

$5 x - 4 y + 2 z = 11$

Multiply $- 2$ by $- 1$ .

$- x + 5 y - 4 + 2 x - (3 y) = - 1$

$z = 4 - 2 x + 3 y$

$5 x - 4 y + 2 z = 11$

Multiply $3$ by $- 1$ .

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

$z = 4 - 2 x + 3 y$

$5 x - 4 y + 2 z = 11$

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

$z = 4 - 2 x + 3 y$

$5 x - 4 y + 2 z = 11$

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

$z = 4 - 2 x + 3 y$

$5 x - 4 y + 2 z = 11$

Simplify by adding terms.

Add $- x$ and $2 x$ .

$x + 5 y - 4 - 3 y = - 1$

$z = 4 - 2 x + 3 y$

$5 x - 4 y + 2 z = 11$

Subtract $3 y$ from $5 y$ .

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

$5 x - 4 y + 2 z = 11$

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

$5 x - 4 y + 2 z = 11$

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

$5 x - 4 y + 2 z = 11$

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

$5 x - 4 y + 2 z = 11$

Replace all occurrences of $z$ in $5 x - 4 y + 2 z = 11$ with $4 - 2 x + 3 y$ .

$5 x - 4 y + 2 (4 - 2 x + 3 y) = 11$

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

Simplify the left side.

Simplify $5 x - 4 y + 2 (4 - 2 x + 3 y)$ .

Simplify each term.

Apply the distributive property.

$5 x - 4 y + 2 \cdot 4 + 2 (- 2 x) + 2 (3 y) = 11$

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

Simplify.

Multiply $2$ by $4$ .

$5 x - 4 y + 8 + 2 (- 2 x) + 2 (3 y) = 11$

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

Multiply $- 2$ by $2$ .

$5 x - 4 y + 8 - 4 x + 2 (3 y) = 11$

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

Multiply $3$ by $2$ .

$5 x - 4 y + 8 - 4 x + 6 y = 11$

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

$5 x - 4 y + 8 - 4 x + 6 y = 11$

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

$5 x - 4 y + 8 - 4 x + 6 y = 11$

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

Simplify by adding terms.

Subtract $4 x$ from $5 x$ .

$x - 4 y + 8 + 6 y = 11$

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

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

$x + 2 y + 8 = 11$

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

$x + 2 y + 8 = 11$

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

$x + 2 y + 8 = 11$

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

$x + 2 y + 8 = 11$

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

$x + 2 y + 8 = 11$

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

Step 3

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

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

$x + 8 = 11 - 2 y$

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

Subtract $8$ from both sides of the equation.

$x = 11 - 2 y - 8$

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

Subtract $8$ from $11$ .

$x = - 2 y + 3$

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

$x = - 2 y + 3$

$x + 2 y - 4 = - 1$

$z = 4 - 2 x + 3 y$

Step 4

Replace all occurrences of $x$ with $- 2 y + 3$ in each equation.

Replace all occurrences of $x$ in $x + 2 y - 4 = - 1$ with $- 2 y + 3$ .

$(- 2 y + 3) + 2 y - 4 = - 1$

$x = - 2 y + 3$

$z = 4 - 2 x + 3 y$

Simplify the left side.

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

Combine the opposite terms in $(- 2 y + 3) + 2 y - 4$ .

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

$0 + 3 - 4 = - 1$

$x = - 2 y + 3$

$z = 4 - 2 x + 3 y$

Add $0$ and $3$ .

$3 - 4 = - 1$

$x = - 2 y + 3$

$z = 4 - 2 x + 3 y$

$3 - 4 = - 1$

$x = - 2 y + 3$

$z = 4 - 2 x + 3 y$

Subtract $4$ from $3$ .

$- 1 = - 1$

$x = - 2 y + 3$

$z = 4 - 2 x + 3 y$

$- 1 = - 1$

$x = - 2 y + 3$

$z = 4 - 2 x + 3 y$

$- 1 = - 1$

$x = - 2 y + 3$

$z = 4 - 2 x + 3 y$

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

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

$- 1 = - 1$

$x = - 2 y + 3$

Simplify the right side.

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

Simplify each term.

Apply the distributive property.

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

$- 1 = - 1$

$x = - 2 y + 3$

Multiply $- 2$ by $- 2$ .

$z = 4 + 4 y - 2 \cdot 3 + 3 y$

$- 1 = - 1$

$x = - 2 y + 3$

Multiply $- 2$ by $3$ .

$z = 4 + 4 y - 6 + 3 y$

$- 1 = - 1$

$x = - 2 y + 3$

$z = 4 + 4 y - 6 + 3 y$

$- 1 = - 1$

$x = - 2 y + 3$

Simplify by adding terms.

Subtract $6$ from $4$ .

$z = 4 y - 2 + 3 y$

$- 1 = - 1$

$x = - 2 y + 3$

Add $4 y$ and $3 y$ .

$z = 7 y - 2$

$- 1 = - 1$

$x = - 2 y + 3$

$z = 7 y - 2$

$- 1 = - 1$

$x = - 2 y + 3$

$z = 7 y - 2$

$- 1 = - 1$

$x = - 2 y + 3$

$z = 7 y - 2$

$- 1 = - 1$

$x = - 2 y + 3$

$z = 7 y - 2$

$- 1 = - 1$

$x = - 2 y + 3$

Step 5

Remove any equations from the system that are always true.

$z = 7 y - 2$

$x = - 2 y + 3$

Solve the System of Equations 2x-3y+z=4 -x+5y-z=-1 5x-4y+2z=11

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