Solve by Substitution x-3y-z=9 , -x+8y-4z=-24 , 2x-15y+7z=45

$x - 3 y - z = 9$ , $- x + 8 y - 4 z = - 24$ , $2 x - 15 y + 7 z = 45$

Step 1

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

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

$x - z = 9 + 3 y$

$- x + 8 y - 4 z = - 24$

$2 x - 15 y + 7 z = 45$

Add $z$ to both sides of the equation.

$x = 9 + 3 y + z$

$- x + 8 y - 4 z = - 24$

$2 x - 15 y + 7 z = 45$

$x = 9 + 3 y + z$

$- x + 8 y - 4 z = - 24$

$2 x - 15 y + 7 z = 45$

Step 2

Replace all occurrences of $x$ with $9 + 3 y + z$ in each equation.

Replace all occurrences of $x$ in $- x + 8 y - 4 z = - 24$ with $9 + 3 y + z$ .

$- (9 + 3 y + z) + 8 y - 4 z = - 24$

$x = 9 + 3 y + z$

$2 x - 15 y + 7 z = 45$

Simplify the left side.

Simplify $- (9 + 3 y + z) + 8 y - 4 z$ .

Simplify each term.

Apply the distributive property.

$- 1 \cdot 9 - (3 y) - z + 8 y - 4 z = - 24$

$x = 9 + 3 y + z$

$2 x - 15 y + 7 z = 45$

Simplify.

Multiply $- 1$ by $9$ .

$- 9 - (3 y) - z + 8 y - 4 z = - 24$

$x = 9 + 3 y + z$

$2 x - 15 y + 7 z = 45$

Multiply $3$ by $- 1$ .

$- 9 - 3 y - z + 8 y - 4 z = - 24$

$x = 9 + 3 y + z$

$2 x - 15 y + 7 z = 45$

$- 9 - 3 y - z + 8 y - 4 z = - 24$

$x = 9 + 3 y + z$

$2 x - 15 y + 7 z = 45$

$- 9 - 3 y - z + 8 y - 4 z = - 24$

$x = 9 + 3 y + z$

$2 x - 15 y + 7 z = 45$

Simplify by adding terms.

Add $- 3 y$ and $8 y$ .

$- 9 + 5 y - z - 4 z = - 24$

$x = 9 + 3 y + z$

$2 x - 15 y + 7 z = 45$

Subtract $4 z$ from $- z$ .

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

$2 x - 15 y + 7 z = 45$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

$2 x - 15 y + 7 z = 45$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

$2 x - 15 y + 7 z = 45$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

$2 x - 15 y + 7 z = 45$

Replace all occurrences of $x$ in $2 x - 15 y + 7 z = 45$ with $9 + 3 y + z$ .

$2 (9 + 3 y + z) - 15 y + 7 z = 45$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

Simplify the left side.

Simplify $2 (9 + 3 y + z) - 15 y + 7 z$ .

Simplify each term.

Apply the distributive property.

$2 \cdot 9 + 2 (3 y) + 2 z - 15 y + 7 z = 45$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

Simplify.

Multiply $2$ by $9$ .

$18 + 2 (3 y) + 2 z - 15 y + 7 z = 45$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

Multiply $3$ by $2$ .

$18 + 6 y + 2 z - 15 y + 7 z = 45$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

$18 + 6 y + 2 z - 15 y + 7 z = 45$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

$18 + 6 y + 2 z - 15 y + 7 z = 45$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

Simplify by adding terms.

Subtract $15 y$ from $6 y$ .

$18 - 9 y + 2 z + 7 z = 45$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

Add $2 z$ and $7 z$ .

$18 - 9 y + 9 z = 45$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

$18 - 9 y + 9 z = 45$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

$18 - 9 y + 9 z = 45$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

$18 - 9 y + 9 z = 45$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

$18 - 9 y + 9 z = 45$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

Step 3

Solve for $y$ in $18 - 9 y + 9 z = 45$ .

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

Subtract $18$ from both sides of the equation.

$- 9 y + 9 z = 45 - 18$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

Subtract $9 z$ from both sides of the equation.

$- 9 y = 45 - 18 - 9 z$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

Subtract $18$ from $45$ .

$- 9 y = 27 - 9 z$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

$- 9 y = 27 - 9 z$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

Divide each term in $- 9 y = 27 - 9 z$ by $- 9$ and simplify.

Divide each term in $- 9 y = 27 - 9 z$ by $- 9$ .

$\frac{- 9 y}{- 9} = \frac{27}{- 9} + \frac{- 9 z}{- 9}$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 y}{- 9} = \frac{27}{- 9} + \frac{- 9 z}{- 9}$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

Divide $y$ by $1$ .

$y = \frac{27}{- 9} + \frac{- 9 z}{- 9}$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

$y = \frac{27}{- 9} + \frac{- 9 z}{- 9}$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

$y = \frac{27}{- 9} + \frac{- 9 z}{- 9}$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

Simplify the right side.

Simplify each term.

Divide $27$ by $- 9$ .

$y = - 3 + \frac{- 9 z}{- 9}$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

Cancel the common factor of $- 9$ .

Cancel the common factor.

$y = - 3 + \frac{- 9 z}{- 9}$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

Divide $z$ by $1$ .

$y = - 3 + z$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

$y = - 3 + z$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

$y = - 3 + z$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

$y = - 3 + z$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

$y = - 3 + z$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

$y = - 3 + z$

$- 9 + 5 y - 5 z = - 24$

$x = 9 + 3 y + z$

Step 4

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

Replace all occurrences of $y$ in $- 9 + 5 y - 5 z = - 24$ with $- 3 + z$ .

$- 9 + 5 (- 3 + z) - 5 z = - 24$

$y = - 3 + z$

$x = 9 + 3 y + z$

Simplify the left side.

Simplify $- 9 + 5 (- 3 + z) - 5 z$ .

Simplify each term.

Apply the distributive property.

$- 9 + 5 \cdot - 3 + 5 z - 5 z = - 24$

$y = - 3 + z$

$x = 9 + 3 y + z$

Multiply $5$ by $- 3$ .

$- 9 - 15 + 5 z - 5 z = - 24$

$y = - 3 + z$

$x = 9 + 3 y + z$

$- 9 - 15 + 5 z - 5 z = - 24$

$y = - 3 + z$

$x = 9 + 3 y + z$

Simplify by adding terms.

Combine the opposite terms in $- 9 - 15 + 5 z - 5 z$ .

Subtract $5 z$ from $5 z$ .

$- 9 - 15 + 0 = - 24$

$y = - 3 + z$

$x = 9 + 3 y + z$

Add $- 9 - 15$ and $0$ .

$- 9 - 15 = - 24$

$y = - 3 + z$

$x = 9 + 3 y + z$

$- 9 - 15 = - 24$

$y = - 3 + z$

$x = 9 + 3 y + z$

Subtract $15$ from $- 9$ .

$- 24 = - 24$

$y = - 3 + z$

$x = 9 + 3 y + z$

$- 24 = - 24$

$y = - 3 + z$

$x = 9 + 3 y + z$

$- 24 = - 24$

$y = - 3 + z$

$x = 9 + 3 y + z$

$- 24 = - 24$

$y = - 3 + z$

$x = 9 + 3 y + z$

Replace all occurrences of $y$ in $x = 9 + 3 y + z$ with $- 3 + z$ .

$x = 9 + 3 (- 3 + z) + z$

$- 24 = - 24$

$y = - 3 + z$

Simplify the right side.

Simplify $9 + 3 (- 3 + z) + z$ .

Simplify each term.

Apply the distributive property.

$x = 9 + 3 \cdot - 3 + 3 z + z$

$- 24 = - 24$

$y = - 3 + z$

Multiply $3$ by $- 3$ .

$x = 9 - 9 + 3 z + z$

$- 24 = - 24$

$y = - 3 + z$

$x = 9 - 9 + 3 z + z$

$- 24 = - 24$

$y = - 3 + z$

Simplify by adding terms.

Subtract $9$ from $9$ .

$x = 0 + 3 z + z$

$- 24 = - 24$

$y = - 3 + z$

Add $0$ and $3 z$ .

$x = 3 z + z$

$- 24 = - 24$

$y = - 3 + z$

Add $3 z$ and $z$ .

$x = 4 z$

$- 24 = - 24$

$y = - 3 + z$

$x = 4 z$

$- 24 = - 24$

$y = - 3 + z$

$x = 4 z$

$- 24 = - 24$

$y = - 3 + z$

$x = 4 z$

$- 24 = - 24$

$y = - 3 + z$

$x = 4 z$

$- 24 = - 24$

$y = - 3 + z$

Step 5

Remove any equations from the system that are always true.

$x = 4 z$

$y = - 3 + z$

Solve by Substitution x-3y-z=9 , -x+8y-4z=-24 , 2x-15y+7z=45

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