Solve the System of Equations 2x-y+3z=-5 -x+y-z=1 4x-3y+5z=-7

$2 x - y + 3 z = - 5$ $- x + y - z = 1$ $4 x - 3 y + 5 z = - 7$

Step 1

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

Add $x$ to both sides of the equation.

$y - z = 1 + x$

$2 x - y + 3 z = - 5$

$4 x - 3 y + 5 z = - 7$

Add $z$ to both sides of the equation.

$y = 1 + x + z$

$2 x - y + 3 z = - 5$

$4 x - 3 y + 5 z = - 7$

$y = 1 + x + z$

$2 x - y + 3 z = - 5$

$4 x - 3 y + 5 z = - 7$

Step 2

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

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

$2 x - (1 + x + z) + 3 z = - 5$

$y = 1 + x + z$

$4 x - 3 y + 5 z = - 7$

Simplify the left side.

Simplify $2 x - (1 + x + z) + 3 z$ .

Simplify each term.

Apply the distributive property.

$2 x - 1 \cdot 1 - x - z + 3 z = - 5$

$y = 1 + x + z$

$4 x - 3 y + 5 z = - 7$

Multiply $- 1$ by $1$ .

$2 x - 1 - x - z + 3 z = - 5$

$y = 1 + x + z$

$4 x - 3 y + 5 z = - 7$

$2 x - 1 - x - z + 3 z = - 5$

$y = 1 + x + z$

$4 x - 3 y + 5 z = - 7$

Simplify by adding terms.

Subtract $x$ from $2 x$ .

$x - 1 - z + 3 z = - 5$

$y = 1 + x + z$

$4 x - 3 y + 5 z = - 7$

Add $- z$ and $3 z$ .

$x - 1 + 2 z = - 5$

$y = 1 + x + z$

$4 x - 3 y + 5 z = - 7$

$x - 1 + 2 z = - 5$

$y = 1 + x + z$

$4 x - 3 y + 5 z = - 7$

$x - 1 + 2 z = - 5$

$y = 1 + x + z$

$4 x - 3 y + 5 z = - 7$

$x - 1 + 2 z = - 5$

$y = 1 + x + z$

$4 x - 3 y + 5 z = - 7$

Replace all occurrences of $y$ in $4 x - 3 y + 5 z = - 7$ with $1 + x + z$ .

$4 x - 3 (1 + x + z) + 5 z = - 7$

$x - 1 + 2 z = - 5$

$y = 1 + x + z$

Simplify the left side.

Simplify $4 x - 3 (1 + x + z) + 5 z$ .

Simplify each term.

Apply the distributive property.

$4 x - 3 \cdot 1 - 3 x - 3 z + 5 z = - 7$

$x - 1 + 2 z = - 5$

$y = 1 + x + z$

Multiply $- 3$ by $1$ .

$4 x - 3 - 3 x - 3 z + 5 z = - 7$

$x - 1 + 2 z = - 5$

$y = 1 + x + z$

$4 x - 3 - 3 x - 3 z + 5 z = - 7$

$x - 1 + 2 z = - 5$

$y = 1 + x + z$

Simplify by adding terms.

Subtract $3 x$ from $4 x$ .

$x - 3 - 3 z + 5 z = - 7$

$x - 1 + 2 z = - 5$

$y = 1 + x + z$

Add $- 3 z$ and $5 z$ .

$x - 3 + 2 z = - 7$

$x - 1 + 2 z = - 5$

$y = 1 + x + z$

$x - 3 + 2 z = - 7$

$x - 1 + 2 z = - 5$

$y = 1 + x + z$

$x - 3 + 2 z = - 7$

$x - 1 + 2 z = - 5$

$y = 1 + x + z$

$x - 3 + 2 z = - 7$

$x - 1 + 2 z = - 5$

$y = 1 + x + z$

$x - 3 + 2 z = - 7$

$x - 1 + 2 z = - 5$

$y = 1 + x + z$

Step 3

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

Add $3$ to both sides of the equation.

$x + 2 z = - 7 + 3$

$x - 1 + 2 z = - 5$

$y = 1 + x + z$

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

$x = - 7 + 3 - 2 z$

$x - 1 + 2 z = - 5$

$y = 1 + x + z$

Add $- 7$ and $3$ .

$x = - 4 - 2 z$

$x - 1 + 2 z = - 5$

$y = 1 + x + z$

$x = - 4 - 2 z$

$x - 1 + 2 z = - 5$

$y = 1 + x + z$

Step 4

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

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

$(- 4 - 2 z) - 1 + 2 z = - 5$

$x = - 4 - 2 z$

$y = 1 + x + z$

Simplify the left side.

Simplify $(- 4 - 2 z) - 1 + 2 z$ .

Combine the opposite terms in $(- 4 - 2 z) - 1 + 2 z$ .

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

$- 4 + 0 - 1 = - 5$

$x = - 4 - 2 z$

$y = 1 + x + z$

Add $- 4$ and $0$ .

$- 4 - 1 = - 5$

$x = - 4 - 2 z$

$y = 1 + x + z$

$- 4 - 1 = - 5$

$x = - 4 - 2 z$

$y = 1 + x + z$

Subtract $1$ from $- 4$ .

$- 5 = - 5$

$x = - 4 - 2 z$

$y = 1 + x + z$

$- 5 = - 5$

$x = - 4 - 2 z$

$y = 1 + x + z$

$- 5 = - 5$

$x = - 4 - 2 z$

$y = 1 + x + z$

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

$y = 1 - 4 - 2 z + z$

$- 5 = - 5$

$x = - 4 - 2 z$

Simplify $y = 1 - 4 - 2 z + z$ .

Simplify the left side.

Remove parentheses.

$y = 1 - 4 - 2 z + z$

$- 5 = - 5$

$x = - 4 - 2 z$

$y = 1 - 4 - 2 z + z$

$- 5 = - 5$

$x = - 4 - 2 z$

Simplify the right side.

Simplify $1 - 4 - 2 z + z$ .

Subtract $4$ from $1$ .

$y = - 3 - 2 z + z$

$- 5 = - 5$

$x = - 4 - 2 z$

Add $- 2 z$ and $z$ .

$y = - 3 - z$

$- 5 = - 5$

$x = - 4 - 2 z$

$y = - 3 - z$

$- 5 = - 5$

$x = - 4 - 2 z$

$y = - 3 - z$

$- 5 = - 5$

$x = - 4 - 2 z$

$y = - 3 - z$

$- 5 = - 5$

$x = - 4 - 2 z$

$y = - 3 - z$

$- 5 = - 5$

$x = - 4 - 2 z$

Step 5

Remove any equations from the system that are always true.

$y = - 3 - z$

$x = - 4 - 2 z$

Solve the System of Equations 2x-y+3z=-5 -x+y-z=1 4x-3y+5z=-7

Related Questions

Support LCMGCF.com by disabling your adblocker.