Algebra Examples

$2 x - 4 y - z = 10$ $4 x - 8 y - 2 z = 16$ $3 x + y + z = 12$

Step 1

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

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Step 1.1

Subtract $3 x$ from both sides of the equation.

$y + z = 12 - 3 x$

$2 x - 4 y - z = 10$

$4 x - 8 y - 2 z = 16$

Step 1.2

Subtract $z$ from both sides of the equation.

$y = 12 - 3 x - z$

$2 x - 4 y - z = 10$

$4 x - 8 y - 2 z = 16$

$y = 12 - 3 x - z$

$2 x - 4 y - z = 10$

$4 x - 8 y - 2 z = 16$

Step 2

Replace all occurrences of $y$ with $12 - 3 x - z$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $2 x - 4 y - z = 10$ with $12 - 3 x - z$ .

$2 x - 4 (12 - 3 x - z) - z = 10$

$y = 12 - 3 x - z$

$4 x - 8 y - 2 z = 16$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 x - 4 (12 - 3 x - z) - z$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$2 x - 4 \cdot 12 - 4 (- 3 x) - 4 (- z) - z = 10$

$y = 12 - 3 x - z$

$4 x - 8 y - 2 z = 16$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 4$ by $12$ .

$2 x - 48 - 4 (- 3 x) - 4 (- z) - z = 10$

$y = 12 - 3 x - z$

$4 x - 8 y - 2 z = 16$

Step 2.2.1.1.2.2

Multiply $- 3$ by $- 4$ .

$2 x - 48 + 12 x - 4 (- z) - z = 10$

$y = 12 - 3 x - z$

$4 x - 8 y - 2 z = 16$

Step 2.2.1.1.2.3

Multiply $- 1$ by $- 4$ .

$2 x - 48 + 12 x + 4 z - z = 10$

$y = 12 - 3 x - z$

$4 x - 8 y - 2 z = 16$

$2 x - 48 + 12 x + 4 z - z = 10$

$y = 12 - 3 x - z$

$4 x - 8 y - 2 z = 16$

$2 x - 48 + 12 x + 4 z - z = 10$

$y = 12 - 3 x - z$

$4 x - 8 y - 2 z = 16$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $2 x$ and $12 x$ .

$14 x - 48 + 4 z - z = 10$

$y = 12 - 3 x - z$

$4 x - 8 y - 2 z = 16$

Step 2.2.1.2.2

Subtract $z$ from $4 z$ .

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

$4 x - 8 y - 2 z = 16$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

$4 x - 8 y - 2 z = 16$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

$4 x - 8 y - 2 z = 16$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

$4 x - 8 y - 2 z = 16$

Step 2.3

Replace all occurrences of $y$ in $4 x - 8 y - 2 z = 16$ with $12 - 3 x - z$ .

$4 x - 8 (12 - 3 x - z) - 2 z = 16$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $4 x - 8 (12 - 3 x - z) - 2 z$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Apply the distributive property.

$4 x - 8 \cdot 12 - 8 (- 3 x) - 8 (- z) - 2 z = 16$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 8$ by $12$ .

$4 x - 96 - 8 (- 3 x) - 8 (- z) - 2 z = 16$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

Step 2.4.1.1.2.2

Multiply $- 3$ by $- 8$ .

$4 x - 96 + 24 x - 8 (- z) - 2 z = 16$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

Step 2.4.1.1.2.3

Multiply $- 1$ by $- 8$ .

$4 x - 96 + 24 x + 8 z - 2 z = 16$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

$4 x - 96 + 24 x + 8 z - 2 z = 16$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

$4 x - 96 + 24 x + 8 z - 2 z = 16$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Add $4 x$ and $24 x$ .

$28 x - 96 + 8 z - 2 z = 16$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

Step 2.4.1.2.2

Subtract $2 z$ from $8 z$ .

$28 x - 96 + 6 z = 16$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

$28 x - 96 + 6 z = 16$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

$28 x - 96 + 6 z = 16$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

$28 x - 96 + 6 z = 16$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

$28 x - 96 + 6 z = 16$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

Step 3

Solve for $x$ in $28 x - 96 + 6 z = 16$ .

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Step 3.1

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

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Step 3.1.1

Add $96$ to both sides of the equation.

$28 x + 6 z = 16 + 96$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

Step 3.1.2

Subtract $6 z$ from both sides of the equation.

$28 x = 16 + 96 - 6 z$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

Step 3.1.3

Add $16$ and $96$ .

$28 x = 112 - 6 z$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

$28 x = 112 - 6 z$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

Step 3.2

Divide each term in $28 x = 112 - 6 z$ by $28$ and simplify.

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Step 3.2.1

Divide each term in $28 x = 112 - 6 z$ by $28$ .

$\frac{28 x}{28} = \frac{112}{28} + \frac{- 6 z}{28}$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $28$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{28 x}{28} = \frac{112}{28} + \frac{- 6 z}{28}$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{112}{28} + \frac{- 6 z}{28}$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

$x = \frac{112}{28} + \frac{- 6 z}{28}$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

$x = \frac{112}{28} + \frac{- 6 z}{28}$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Divide $112$ by $28$ .

$x = 4 + \frac{- 6 z}{28}$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

Step 3.2.3.1.2

Cancel the common factor of $- 6$ and $28$ .

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Step 3.2.3.1.2.1

Factor $2$ out of $- 6 z$ .

$x = 4 + \frac{2 (- 3 z)}{28}$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

Step 3.2.3.1.2.2

Cancel the common factors.

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Step 3.2.3.1.2.2.1

Factor $2$ out of $28$ .

$x = 4 + \frac{2 (- 3 z)}{2 (14)}$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

Step 3.2.3.1.2.2.2

Cancel the common factor.

$x = 4 + \frac{2 (- 3 z)}{2 \cdot 14}$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

Step 3.2.3.1.2.2.3

Rewrite the expression.

$x = 4 + \frac{- 3 z}{14}$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

$x = 4 + \frac{- 3 z}{14}$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

$x = 4 + \frac{- 3 z}{14}$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

Step 3.2.3.1.3

Move the negative in front of the fraction.

$x = 4 - \frac{3 z}{14}$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

$x = 4 - \frac{3 z}{14}$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

$x = 4 - \frac{3 z}{14}$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

$x = 4 - \frac{3 z}{14}$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

$x = 4 - \frac{3 z}{14}$

$14 x - 48 + 3 z = 10$

$y = 12 - 3 x - z$

Step 4

Replace all occurrences of $x$ with $4 - \frac{3 z}{14}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $14 x - 48 + 3 z = 10$ with $4 - \frac{3 z}{14}$ .

$14 (4 - \frac{3 z}{14}) - 48 + 3 z = 10$

$x = 4 - \frac{3 z}{14}$

$y = 12 - 3 x - z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $14 (4 - \frac{3 z}{14}) - 48 + 3 z$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$14 \cdot 4 + 14 (- \frac{3 z}{14}) - 48 + 3 z = 10$

$x = 4 - \frac{3 z}{14}$

$y = 12 - 3 x - z$

Step 4.2.1.1.2

Multiply $14$ by $4$ .

$56 + 14 (- \frac{3 z}{14}) - 48 + 3 z = 10$

$x = 4 - \frac{3 z}{14}$

$y = 12 - 3 x - z$

Step 4.2.1.1.3

Cancel the common factor of $14$ .

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Step 4.2.1.1.3.1

Move the leading negative in $- \frac{3 z}{14}$ into the numerator.

$56 + 14 (\frac{- 3 z}{14}) - 48 + 3 z = 10$

$x = 4 - \frac{3 z}{14}$

$y = 12 - 3 x - z$

Step 4.2.1.1.3.2

Cancel the common factor.

$56 + 14 (\frac{- 3 z}{14}) - 48 + 3 z = 10$

$x = 4 - \frac{3 z}{14}$

$y = 12 - 3 x - z$

Step 4.2.1.1.3.3

Rewrite the expression.

$56 - 3 z - 48 + 3 z = 10$

$x = 4 - \frac{3 z}{14}$

$y = 12 - 3 x - z$

$56 - 3 z - 48 + 3 z = 10$

$x = 4 - \frac{3 z}{14}$

$y = 12 - 3 x - z$

$56 - 3 z - 48 + 3 z = 10$

$x = 4 - \frac{3 z}{14}$

$y = 12 - 3 x - z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Combine the opposite terms in $56 - 3 z - 48 + 3 z$ .

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Step 4.2.1.2.1.1

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

$56 + 0 - 48 = 10$

$x = 4 - \frac{3 z}{14}$

$y = 12 - 3 x - z$

Step 4.2.1.2.1.2

Add $56$ and $0$ .

$56 - 48 = 10$

$x = 4 - \frac{3 z}{14}$

$y = 12 - 3 x - z$

$56 - 48 = 10$

$x = 4 - \frac{3 z}{14}$

$y = 12 - 3 x - z$

Step 4.2.1.2.2

Subtract $48$ from $56$ .

$8 = 10$

$x = 4 - \frac{3 z}{14}$

$y = 12 - 3 x - z$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

$y = 12 - 3 x - z$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

$y = 12 - 3 x - z$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

$y = 12 - 3 x - z$

Step 4.3

Replace all occurrences of $x$ in $y = 12 - 3 x - z$ with $4 - \frac{3 z}{14}$ .

$y = 12 - 3 (4 - \frac{3 z}{14}) - z$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $12 - 3 (4 - \frac{3 z}{14}) - z$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$y = 12 - 3 \cdot 4 - 3 (- \frac{3 z}{14}) - z$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

Step 4.4.1.1.2

Multiply $- 3$ by $4$ .

$y = 12 - 12 - 3 (- \frac{3 z}{14}) - z$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

Step 4.4.1.1.3

Multiply $- 3 (- \frac{3 z}{14})$ .

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Step 4.4.1.1.3.1

Multiply $- 1$ by $- 3$ .

$y = 12 - 12 + 3 (\frac{3 z}{14}) - z$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

Step 4.4.1.1.3.2

Combine $3$ and $\frac{3 z}{14}$ .

$y = 12 - 12 + \frac{3 (3 z)}{14} - z$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

Step 4.4.1.1.3.3

Multiply $3$ by $3$ .

$y = 12 - 12 + \frac{9 z}{14} - z$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

$y = 12 - 12 + \frac{9 z}{14} - z$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

$y = 12 - 12 + \frac{9 z}{14} - z$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

Step 4.4.1.2

Simplify by subtracting numbers.

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Step 4.4.1.2.1

Subtract $12$ from $12$ .

$y = 0 + \frac{9 z}{14} - z$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

Step 4.4.1.2.2

Add $0$ and $\frac{9 z}{14}$ .

$y = \frac{9 z}{14} - z$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

$y = \frac{9 z}{14} - z$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

Step 4.4.1.3

To write $- z$ as a fraction with a common denominator, multiply by $\frac{14}{14}$ .

$y = \frac{9 z}{14} - z \cdot \frac{14}{14}$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

Step 4.4.1.4

Simplify terms.

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Step 4.4.1.4.1

Combine $- z$ and $\frac{14}{14}$ .

$y = \frac{9 z}{14} + \frac{- z \cdot 14}{14}$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

Step 4.4.1.4.2

Combine the numerators over the common denominator.

$y = \frac{9 z - z \cdot 14}{14}$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

$y = \frac{9 z - z \cdot 14}{14}$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

Step 4.4.1.5

Simplify the numerator.

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Step 4.4.1.5.1

Factor $z$ out of $9 z - z \cdot 14$ .

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Step 4.4.1.5.1.1

Factor $z$ out of $9 z$ .

$y = \frac{z \cdot 9 - z \cdot 14}{14}$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

Step 4.4.1.5.1.2

Factor $z$ out of $- z \cdot 14$ .

$y = \frac{z \cdot 9 + z (- 1 \cdot 14)}{14}$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

Step 4.4.1.5.1.3

Factor $z$ out of $z \cdot 9 + z (- 1 \cdot 14)$ .

$y = \frac{z (9 - 1 \cdot 14)}{14}$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

$y = \frac{z (9 - 1 \cdot 14)}{14}$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

Step 4.4.1.5.2

Multiply $- 1$ by $14$ .

$y = \frac{z (9 - 14)}{14}$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

Step 4.4.1.5.3

Subtract $14$ from $9$ .

$y = \frac{z \cdot - 5}{14}$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

$y = \frac{z \cdot - 5}{14}$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

Step 4.4.1.6

Simplify the expression.

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Step 4.4.1.6.1

Move $- 5$ to the left of $z$ .

$y = \frac{- 5 \cdot z}{14}$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

Step 4.4.1.6.2

Move the negative in front of the fraction.

$y = - \frac{5 z}{14}$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

$y = - \frac{5 z}{14}$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

$y = - \frac{5 z}{14}$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

$y = - \frac{5 z}{14}$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

$y = - \frac{5 z}{14}$

$8 = 10$

$x = 4 - \frac{3 z}{14}$

Step 5

Since $8 = 10$ is not true, there is no solution.

No solution