Algebra Examples

$5 x + y - z = 6$ $x + y + z = 2$ $12 x + 4 y = 10$

Step 1

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

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Step 1.1

Subtract $5 x$ from both sides of the equation.

$y - z = 6 - 5 x$

$x + y + z = 2$

$12 x + 4 y = 10$

Step 1.2

Add $z$ to both sides of the equation.

$y = 6 - 5 x + z$

$x + y + z = 2$

$12 x + 4 y = 10$

$y = 6 - 5 x + z$

$x + y + z = 2$

$12 x + 4 y = 10$

Step 2

Replace all occurrences of $y$ with $6 - 5 x + z$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $x + y + z = 2$ with $6 - 5 x + z$ .

$x + 6 - 5 x + z + z = 2$

$y = 6 - 5 x + z$

$12 x + 4 y = 10$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $x + 6 - 5 x + z + z$ .

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Step 2.2.1.1

Remove parentheses.

$x + 6 - 5 x + z + z = 2$

$y = 6 - 5 x + z$

$12 x + 4 y = 10$

Step 2.2.1.2

Subtract $5 x$ from $x$ .

$- 4 x + 6 + z + z = 2$

$y = 6 - 5 x + z$

$12 x + 4 y = 10$

Step 2.2.1.3

Add $z$ and $z$ .

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

$12 x + 4 y = 10$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

$12 x + 4 y = 10$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

$12 x + 4 y = 10$

Step 2.3

Replace all occurrences of $y$ in $12 x + 4 y = 10$ with $6 - 5 x + z$ .

$12 x + 4 (6 - 5 x + z) = 10$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $12 x + 4 (6 - 5 x + z)$ .

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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.

$12 x + 4 \cdot 6 + 4 (- 5 x) + 4 z = 10$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $4$ by $6$ .

$12 x + 24 + 4 (- 5 x) + 4 z = 10$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

Step 2.4.1.1.2.2

Multiply $- 5$ by $4$ .

$12 x + 24 - 20 x + 4 z = 10$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

$12 x + 24 - 20 x + 4 z = 10$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

$12 x + 24 - 20 x + 4 z = 10$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

Step 2.4.1.2

Subtract $20 x$ from $12 x$ .

$- 8 x + 24 + 4 z = 10$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

$- 8 x + 24 + 4 z = 10$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

$- 8 x + 24 + 4 z = 10$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

$- 8 x + 24 + 4 z = 10$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

Step 3

Solve for $x$ in $- 8 x + 24 + 4 z = 10$ .

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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

Subtract $24$ from both sides of the equation.

$- 8 x + 4 z = 10 - 24$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

Step 3.1.2

Subtract $4 z$ from both sides of the equation.

$- 8 x = 10 - 24 - 4 z$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

Step 3.1.3

Subtract $24$ from $10$ .

$- 8 x = - 14 - 4 z$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

$- 8 x = - 14 - 4 z$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

Step 3.2

Divide each term in $- 8 x = - 14 - 4 z$ by $- 8$ and simplify.

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Step 3.2.1

Divide each term in $- 8 x = - 14 - 4 z$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{- 14}{- 8} + \frac{- 4 z}{- 8}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 8$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{- 14}{- 8} + \frac{- 4 z}{- 8}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 14}{- 8} + \frac{- 4 z}{- 8}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

$x = \frac{- 14}{- 8} + \frac{- 4 z}{- 8}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

$x = \frac{- 14}{- 8} + \frac{- 4 z}{- 8}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

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

Cancel the common factor of $- 14$ and $- 8$ .

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Step 3.2.3.1.1.1

Factor $- 2$ out of $- 14$ .

$x = \frac{- 2 \cdot 7}{- 8} + \frac{- 4 z}{- 8}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

Step 3.2.3.1.1.2

Cancel the common factors.

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Step 3.2.3.1.1.2.1

Factor $- 2$ out of $- 8$ .

$x = \frac{- 2 \cdot 7}{- 2 \cdot 4} + \frac{- 4 z}{- 8}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

Step 3.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{- 2 \cdot 7}{- 2 \cdot 4} + \frac{- 4 z}{- 8}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

Step 3.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{7}{4} + \frac{- 4 z}{- 8}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

$x = \frac{7}{4} + \frac{- 4 z}{- 8}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

$x = \frac{7}{4} + \frac{- 4 z}{- 8}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

Step 3.2.3.1.2

Cancel the common factor of $- 4$ and $- 8$ .

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Step 3.2.3.1.2.1

Factor $- 4$ out of $- 4 z$ .

$x = \frac{7}{4} + \frac{- 4 z}{- 8}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

Step 3.2.3.1.2.2

Cancel the common factors.

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Step 3.2.3.1.2.2.1

Factor $- 4$ out of $- 8$ .

$x = \frac{7}{4} + \frac{- 4 z}{- 4 \cdot 2}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

Step 3.2.3.1.2.2.2

Cancel the common factor.

$x = \frac{7}{4} + \frac{- 4 z}{- 4 \cdot 2}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

Step 3.2.3.1.2.2.3

Rewrite the expression.

$x = \frac{7}{4} + \frac{z}{2}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

$x = \frac{7}{4} + \frac{z}{2}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

$x = \frac{7}{4} + \frac{z}{2}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

$x = \frac{7}{4} + \frac{z}{2}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

$x = \frac{7}{4} + \frac{z}{2}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

$x = \frac{7}{4} + \frac{z}{2}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

$x = \frac{7}{4} + \frac{z}{2}$

$- 4 x + 6 + 2 z = 2$

$y = 6 - 5 x + z$

Step 4

Replace all occurrences of $x$ with $\frac{7}{4} + \frac{z}{2}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $- 4 x + 6 + 2 z = 2$ with $\frac{7}{4} + \frac{z}{2}$ .

$- 4 (\frac{7}{4} + \frac{z}{2}) + 6 + 2 z = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - 5 x + z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 4 (\frac{7}{4} + \frac{z}{2}) + 6 + 2 z$ .

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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.

$- 4 (\frac{7}{4}) - 4 \frac{z}{2} + 6 + 2 z = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - 5 x + z$

Step 4.2.1.1.2

Cancel the common factor of $4$ .

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Step 4.2.1.1.2.1

Factor $4$ out of $- 4$ .

$4 (- 1) (\frac{7}{4}) - 4 \frac{z}{2} + 6 + 2 z = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - 5 x + z$

Step 4.2.1.1.2.2

Cancel the common factor.

$4 \cdot (- 1 (\frac{7}{4})) - 4 \frac{z}{2} + 6 + 2 z = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - 5 x + z$

Step 4.2.1.1.2.3

Rewrite the expression.

$- 1 \cdot 7 - 4 \frac{z}{2} + 6 + 2 z = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - 5 x + z$

$- 1 \cdot 7 - 4 \frac{z}{2} + 6 + 2 z = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - 5 x + z$

Step 4.2.1.1.3

Multiply $- 1$ by $7$ .

$- 7 - 4 \frac{z}{2} + 6 + 2 z = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - 5 x + z$

Step 4.2.1.1.4

Cancel the common factor of $2$ .

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Step 4.2.1.1.4.1

Factor $2$ out of $- 4$ .

$- 7 + 2 (- 2) (\frac{z}{2}) + 6 + 2 z = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - 5 x + z$

Step 4.2.1.1.4.2

Cancel the common factor.

$- 7 + 2 \cdot (- 2 \frac{z}{2}) + 6 + 2 z = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - 5 x + z$

Step 4.2.1.1.4.3

Rewrite the expression.

$- 7 - 2 z + 6 + 2 z = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - 5 x + z$

$- 7 - 2 z + 6 + 2 z = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - 5 x + z$

$- 7 - 2 z + 6 + 2 z = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - 5 x + z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Combine the opposite terms in $- 7 - 2 z + 6 + 2 z$ .

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Step 4.2.1.2.1.1

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

$- 7 + 0 + 6 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - 5 x + z$

Step 4.2.1.2.1.2

Add $- 7$ and $0$ .

$- 7 + 6 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - 5 x + z$

$- 7 + 6 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - 5 x + z$

Step 4.2.1.2.2

Add $- 7$ and $6$ .

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - 5 x + z$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - 5 x + z$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - 5 x + z$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - 5 x + z$

Step 4.3

Replace all occurrences of $x$ in $y = 6 - 5 x + z$ with $\frac{7}{4} + \frac{z}{2}$ .

$y = 6 - 5 (\frac{7}{4} + \frac{z}{2}) + z$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $6 - 5 (\frac{7}{4} + \frac{z}{2}) + z$ .

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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.

$y = 6 - 5 (\frac{7}{4}) - 5 \frac{z}{2} + z$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

Step 4.4.1.1.2

Multiply $- 5 (\frac{7}{4})$ .

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Step 4.4.1.1.2.1

Combine $- 5$ and $\frac{7}{4}$ .

$y = 6 + \frac{- 5 \cdot 7}{4} - 5 \frac{z}{2} + z$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

Step 4.4.1.1.2.2

Multiply $- 5$ by $7$ .

$y = 6 + \frac{- 35}{4} - 5 \frac{z}{2} + z$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 + \frac{- 35}{4} - 5 \frac{z}{2} + z$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

Step 4.4.1.1.3

Combine $- 5$ and $\frac{z}{2}$ .

$y = 6 + \frac{- 35}{4} + \frac{- 5 z}{2} + z$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

Step 4.4.1.1.4

Simplify each term.

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Step 4.4.1.1.4.1

Move the negative in front of the fraction.

$y = 6 - \frac{35}{4} + \frac{- 5 z}{2} + z$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

Step 4.4.1.1.4.2

Move the negative in front of the fraction.

$y = 6 - \frac{35}{4} - \frac{5 z}{2} + z$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - \frac{35}{4} - \frac{5 z}{2} + z$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = 6 - \frac{35}{4} - \frac{5 z}{2} + z$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

Step 4.4.1.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = 6 \cdot \frac{4}{4} - \frac{35}{4} - \frac{5 z}{2} + z$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

Step 4.4.1.3

Combine $6$ and $\frac{4}{4}$ .

$y = \frac{6 \cdot 4}{4} - \frac{35}{4} - \frac{5 z}{2} + z$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

Step 4.4.1.4

Combine the numerators over the common denominator.

$y = \frac{6 \cdot 4 - 35}{4} - \frac{5 z}{2} + z$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

Step 4.4.1.5

Simplify the numerator.

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Step 4.4.1.5.1

Multiply $6$ by $4$ .

$y = \frac{24 - 35}{4} - \frac{5 z}{2} + z$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

Step 4.4.1.5.2

Subtract $35$ from $24$ .

$y = \frac{- 11}{4} - \frac{5 z}{2} + z$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = \frac{- 11}{4} - \frac{5 z}{2} + z$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

Step 4.4.1.6

Move the negative in front of the fraction.

$y = - \frac{11}{4} - \frac{5 z}{2} + z$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

Step 4.4.1.7

To write $z$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{11}{4} - \frac{5 z}{2} + z \cdot \frac{2}{2}$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

Step 4.4.1.8

Combine $z$ and $\frac{2}{2}$ .

$y = - \frac{11}{4} - \frac{5 z}{2} + \frac{z \cdot 2}{2}$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

Step 4.4.1.9

Combine the numerators over the common denominator.

$y = - \frac{11}{4} + \frac{- 5 z + z \cdot 2}{2}$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

Step 4.4.1.10

Add $- 5 z$ and $z \cdot 2$ .

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Step 4.4.1.10.1

Reorder $z$ and $2$ .

$y = - \frac{11}{4} + \frac{- 5 z + 2 \cdot z}{2}$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

Step 4.4.1.10.2

Add $- 5 z$ and $2 \cdot z$ .

$y = - \frac{11}{4} + \frac{- 3 z}{2}$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = - \frac{11}{4} + \frac{- 3 z}{2}$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

Step 4.4.1.11

Move the negative in front of the fraction.

$y = - \frac{11}{4} - \frac{3 z}{2}$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = - \frac{11}{4} - \frac{3 z}{2}$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = - \frac{11}{4} - \frac{3 z}{2}$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

$y = - \frac{11}{4} - \frac{3 z}{2}$

$- 1 = 2$

$x = \frac{7}{4} + \frac{z}{2}$

Step 5

Since $- 1 = 2$ is not true, there is no solution.

No solution