Solve by Substitution 2x-3y+z=-3 , x-5y+7z=-11 , -10x+4y-6z=28

$2 x - 3 y + z = - 3$ , $x - 5 y + 7 z = - 11$ , $- 10 x + 4 y - 6 z = 28$

Step 1

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

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

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

$x - 5 y + 7 z = - 11$

$- 10 x + 4 y - 6 z = 28$

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

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

$x - 5 y + 7 z = - 11$

$- 10 x + 4 y - 6 z = 28$

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

$x - 5 y + 7 z = - 11$

$- 10 x + 4 y - 6 z = 28$

Step 2

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

Replace all occurrences of $z$ in $x - 5 y + 7 z = - 11$ with $- 3 - 2 x + 3 y$ .

$x - 5 y + 7 (- 3 - 2 x + 3 y) = - 11$

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

$- 10 x + 4 y - 6 z = 28$

Simplify the left side.

Simplify $x - 5 y + 7 (- 3 - 2 x + 3 y)$ .

Simplify each term.

Apply the distributive property.

$x - 5 y + 7 \cdot - 3 + 7 (- 2 x) + 7 (3 y) = - 11$

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

$- 10 x + 4 y - 6 z = 28$

Simplify.

Multiply $7$ by $- 3$ .

$x - 5 y - 21 + 7 (- 2 x) + 7 (3 y) = - 11$

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

$- 10 x + 4 y - 6 z = 28$

Multiply $- 2$ by $7$ .

$x - 5 y - 21 - 14 x + 7 (3 y) = - 11$

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

$- 10 x + 4 y - 6 z = 28$

Multiply $3$ by $7$ .

$x - 5 y - 21 - 14 x + 21 y = - 11$

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

$- 10 x + 4 y - 6 z = 28$

$x - 5 y - 21 - 14 x + 21 y = - 11$

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

$- 10 x + 4 y - 6 z = 28$

$x - 5 y - 21 - 14 x + 21 y = - 11$

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

$- 10 x + 4 y - 6 z = 28$

Simplify by adding terms.

Subtract $14 x$ from $x$ .

$- 13 x - 5 y - 21 + 21 y = - 11$

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

$- 10 x + 4 y - 6 z = 28$

Add $- 5 y$ and $21 y$ .

$- 13 x + 16 y - 21 = - 11$

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

$- 10 x + 4 y - 6 z = 28$

$- 13 x + 16 y - 21 = - 11$

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

$- 10 x + 4 y - 6 z = 28$

$- 13 x + 16 y - 21 = - 11$

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

$- 10 x + 4 y - 6 z = 28$

$- 13 x + 16 y - 21 = - 11$

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

$- 10 x + 4 y - 6 z = 28$

Replace all occurrences of $z$ in $- 10 x + 4 y - 6 z = 28$ with $- 3 - 2 x + 3 y$ .

$- 10 x + 4 y - 6 (- 3 - 2 x + 3 y) = 28$

$- 13 x + 16 y - 21 = - 11$

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

Simplify the left side.

Simplify $- 10 x + 4 y - 6 (- 3 - 2 x + 3 y)$ .

Simplify each term.

Apply the distributive property.

$- 10 x + 4 y - 6 \cdot - 3 - 6 (- 2 x) - 6 (3 y) = 28$

$- 13 x + 16 y - 21 = - 11$

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

Simplify.

Multiply $- 6$ by $- 3$ .

$- 10 x + 4 y + 18 - 6 (- 2 x) - 6 (3 y) = 28$

$- 13 x + 16 y - 21 = - 11$

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

Multiply $- 2$ by $- 6$ .

$- 10 x + 4 y + 18 + 12 x - 6 (3 y) = 28$

$- 13 x + 16 y - 21 = - 11$

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

Multiply $3$ by $- 6$ .

$- 10 x + 4 y + 18 + 12 x - 18 y = 28$

$- 13 x + 16 y - 21 = - 11$

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

$- 10 x + 4 y + 18 + 12 x - 18 y = 28$

$- 13 x + 16 y - 21 = - 11$

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

$- 10 x + 4 y + 18 + 12 x - 18 y = 28$

$- 13 x + 16 y - 21 = - 11$

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

Simplify by adding terms.

Add $- 10 x$ and $12 x$ .

$2 x + 4 y + 18 - 18 y = 28$

$- 13 x + 16 y - 21 = - 11$

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

Subtract $18 y$ from $4 y$ .

$2 x - 14 y + 18 = 28$

$- 13 x + 16 y - 21 = - 11$

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

$2 x - 14 y + 18 = 28$

$- 13 x + 16 y - 21 = - 11$

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

$2 x - 14 y + 18 = 28$

$- 13 x + 16 y - 21 = - 11$

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

$2 x - 14 y + 18 = 28$

$- 13 x + 16 y - 21 = - 11$

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

$2 x - 14 y + 18 = 28$

$- 13 x + 16 y - 21 = - 11$

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

Step 3

Solve for $x$ in $2 x - 14 y + 18 = 28$ .

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

Add $14 y$ to both sides of the equation.

$2 x + 18 = 28 + 14 y$

$- 13 x + 16 y - 21 = - 11$

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

Subtract $18$ from both sides of the equation.

$2 x = 28 + 14 y - 18$

$- 13 x + 16 y - 21 = - 11$

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

Subtract $18$ from $28$ .

$2 x = 14 y + 10$

$- 13 x + 16 y - 21 = - 11$

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

$2 x = 14 y + 10$

$- 13 x + 16 y - 21 = - 11$

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

Divide each term in $2 x = 14 y + 10$ by $2$ and simplify.

Divide each term in $2 x = 14 y + 10$ by $2$ .

$\frac{2 x}{2} = \frac{14 y}{2} + \frac{10}{2}$

$- 13 x + 16 y - 21 = - 11$

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{14 y}{2} + \frac{10}{2}$

$- 13 x + 16 y - 21 = - 11$

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

Divide $x$ by $1$ .

$x = \frac{14 y}{2} + \frac{10}{2}$

$- 13 x + 16 y - 21 = - 11$

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

$x = \frac{14 y}{2} + \frac{10}{2}$

$- 13 x + 16 y - 21 = - 11$

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

$x = \frac{14 y}{2} + \frac{10}{2}$

$- 13 x + 16 y - 21 = - 11$

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

Simplify the right side.

Simplify each term.

Cancel the common factor of $14$ and $2$ .

Factor $2$ out of $14 y$ .

$x = \frac{2 (7 y)}{2} + \frac{10}{2}$

$- 13 x + 16 y - 21 = - 11$

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

Cancel the common factors.

Factor $2$ out of $2$ .

$x = \frac{2 (7 y)}{2 (1)} + \frac{10}{2}$

$- 13 x + 16 y - 21 = - 11$

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

Cancel the common factor.

$x = \frac{2 (7 y)}{2 \cdot 1} + \frac{10}{2}$

$- 13 x + 16 y - 21 = - 11$

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

Rewrite the expression.

$x = \frac{7 y}{1} + \frac{10}{2}$

$- 13 x + 16 y - 21 = - 11$

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

Divide $7 y$ by $1$ .

$x = 7 y + \frac{10}{2}$

$- 13 x + 16 y - 21 = - 11$

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

$x = 7 y + \frac{10}{2}$

$- 13 x + 16 y - 21 = - 11$

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

$x = 7 y + \frac{10}{2}$

$- 13 x + 16 y - 21 = - 11$

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

Divide $10$ by $2$ .

$x = 7 y + 5$

$- 13 x + 16 y - 21 = - 11$

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

$x = 7 y + 5$

$- 13 x + 16 y - 21 = - 11$

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

$x = 7 y + 5$

$- 13 x + 16 y - 21 = - 11$

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

$x = 7 y + 5$

$- 13 x + 16 y - 21 = - 11$

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

$x = 7 y + 5$

$- 13 x + 16 y - 21 = - 11$

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

Step 4

Replace all occurrences of $x$ with $7 y + 5$ in each equation.

Replace all occurrences of $x$ in $- 13 x + 16 y - 21 = - 11$ with $7 y + 5$ .

$- 13 (7 y + 5) + 16 y - 21 = - 11$

$x = 7 y + 5$

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

Simplify the left side.

Simplify $- 13 (7 y + 5) + 16 y - 21$ .

Simplify each term.

Apply the distributive property.

$- 13 (7 y) - 13 \cdot 5 + 16 y - 21 = - 11$

$x = 7 y + 5$

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

Multiply $7$ by $- 13$ .

$- 91 y - 13 \cdot 5 + 16 y - 21 = - 11$

$x = 7 y + 5$

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

Multiply $- 13$ by $5$ .

$- 91 y - 65 + 16 y - 21 = - 11$

$x = 7 y + 5$

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

$- 91 y - 65 + 16 y - 21 = - 11$

$x = 7 y + 5$

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

Simplify by adding terms.

Add $- 91 y$ and $16 y$ .

$- 75 y - 65 - 21 = - 11$

$x = 7 y + 5$

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

Subtract $21$ from $- 65$ .

$- 75 y - 86 = - 11$

$x = 7 y + 5$

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

$- 75 y - 86 = - 11$

$x = 7 y + 5$

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

$- 75 y - 86 = - 11$

$x = 7 y + 5$

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

$- 75 y - 86 = - 11$

$x = 7 y + 5$

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

Replace all occurrences of $x$ in $z = - 3 - 2 x + 3 y$ with $7 y + 5$ .

$z = - 3 - 2 (7 y + 5) + 3 y$

$- 75 y - 86 = - 11$

$x = 7 y + 5$

Simplify the right side.

Simplify $- 3 - 2 (7 y + 5) + 3 y$ .

Simplify each term.

Apply the distributive property.

$z = - 3 - 2 (7 y) - 2 \cdot 5 + 3 y$

$- 75 y - 86 = - 11$

$x = 7 y + 5$

Multiply $7$ by $- 2$ .

$z = - 3 - 14 y - 2 \cdot 5 + 3 y$

$- 75 y - 86 = - 11$

$x = 7 y + 5$

Multiply $- 2$ by $5$ .

$z = - 3 - 14 y - 10 + 3 y$

$- 75 y - 86 = - 11$

$x = 7 y + 5$

$z = - 3 - 14 y - 10 + 3 y$

$- 75 y - 86 = - 11$

$x = 7 y + 5$

Simplify by adding terms.

Subtract $10$ from $- 3$ .

$z = - 14 y - 13 + 3 y$

$- 75 y - 86 = - 11$

$x = 7 y + 5$

Add $- 14 y$ and $3 y$ .

$z = - 11 y - 13$

$- 75 y - 86 = - 11$

$x = 7 y + 5$

$z = - 11 y - 13$

$- 75 y - 86 = - 11$

$x = 7 y + 5$

$z = - 11 y - 13$

$- 75 y - 86 = - 11$

$x = 7 y + 5$

$z = - 11 y - 13$

$- 75 y - 86 = - 11$

$x = 7 y + 5$

$z = - 11 y - 13$

$- 75 y - 86 = - 11$

$x = 7 y + 5$

Step 5

Solve for $y$ in $- 75 y - 86 = - 11$ .

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

Add $86$ to both sides of the equation.

$- 75 y = - 11 + 86$

$z = - 11 y - 13$

$x = 7 y + 5$

Add $- 11$ and $86$ .

$- 75 y = 75$

$z = - 11 y - 13$

$x = 7 y + 5$

$- 75 y = 75$

$z = - 11 y - 13$

$x = 7 y + 5$

Divide each term in $- 75 y = 75$ by $- 75$ and simplify.

Divide each term in $- 75 y = 75$ by $- 75$ .

$\frac{- 75 y}{- 75} = \frac{75}{- 75}$

$z = - 11 y - 13$

$x = 7 y + 5$

Simplify the left side.

Cancel the common factor of $- 75$ .

Cancel the common factor.

$\frac{- 75 y}{- 75} = \frac{75}{- 75}$

$z = - 11 y - 13$

$x = 7 y + 5$

Divide $y$ by $1$ .

$y = \frac{75}{- 75}$

$z = - 11 y - 13$

$x = 7 y + 5$

$y = \frac{75}{- 75}$

$z = - 11 y - 13$

$x = 7 y + 5$

$y = \frac{75}{- 75}$

$z = - 11 y - 13$

$x = 7 y + 5$

Simplify the right side.

Divide $75$ by $- 75$ .

$y = - 1$

$z = - 11 y - 13$

$x = 7 y + 5$

$y = - 1$

$z = - 11 y - 13$

$x = 7 y + 5$

$y = - 1$

$z = - 11 y - 13$

$x = 7 y + 5$

$y = - 1$

$z = - 11 y - 13$

$x = 7 y + 5$

Step 6

Replace all occurrences of $y$ with $- 1$ in each equation.

Replace all occurrences of $y$ in $z = - 11 y - 13$ with $- 1$ .

$z = - 11 \cdot - 1 - 13$

$y = - 1$

$x = 7 y + 5$

Simplify the right side.

Simplify $- 11 \cdot - 1 - 13$ .

Multiply $- 11$ by $- 1$ .

$z = 11 - 13$

$y = - 1$

$x = 7 y + 5$

Subtract $13$ from $11$ .

$z = - 2$

$y = - 1$

$x = 7 y + 5$

$z = - 2$

$y = - 1$

$x = 7 y + 5$

$z = - 2$

$y = - 1$

$x = 7 y + 5$

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

$x = 7 (- 1) + 5$

$z = - 2$

$y = - 1$

Simplify the right side.

Simplify $7 (- 1) + 5$ .

Multiply $7$ by $- 1$ .

$x = - 7 + 5$

$z = - 2$

$y = - 1$

Add $- 7$ and $5$ .

$x = - 2$

$z = - 2$

$y = - 1$

$x = - 2$

$z = - 2$

$y = - 1$

$x = - 2$

$z = - 2$

$y = - 1$

$x = - 2$

$z = - 2$

$y = - 1$

Step 7

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

$(- 2, - 1, - 2)$

Step 8

The result can be shown in multiple forms.

Point Form:

$(- 2, - 1, - 2)$

Equation Form:

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

Solve by Substitution 2x-3y+z=-3 , x-5y+7z=-11 , -10x+4y-6z=28

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