Algebra Examples

$3 x - 3 y + 6 z = 18$ $x + 2 y - z = 18$ $5 x - 8 y + 13 z = 18$

Step 1

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

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Step 1.1

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

$x - z = 18 - 2 y$

$3 x - 3 y + 6 z = 18$

$5 x - 8 y + 13 z = 18$

Step 1.2

Add $z$ to both sides of the equation.

$x = 18 - 2 y + z$

$3 x - 3 y + 6 z = 18$

$5 x - 8 y + 13 z = 18$

$x = 18 - 2 y + z$

$3 x - 3 y + 6 z = 18$

$5 x - 8 y + 13 z = 18$

Step 2

Replace all occurrences of $x$ with $18 - 2 y + z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 x - 3 y + 6 z = 18$ with $18 - 2 y + z$ .

$3 (18 - 2 y + z) - 3 y + 6 z = 18$

$x = 18 - 2 y + z$

$5 x - 8 y + 13 z = 18$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (18 - 2 y + z) - 3 y + 6 z$ .

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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.

$3 \cdot 18 + 3 (- 2 y) + 3 z - 3 y + 6 z = 18$

$x = 18 - 2 y + z$

$5 x - 8 y + 13 z = 18$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $3$ by $18$ .

$54 + 3 (- 2 y) + 3 z - 3 y + 6 z = 18$

$x = 18 - 2 y + z$

$5 x - 8 y + 13 z = 18$

Step 2.2.1.1.2.2

Multiply $- 2$ by $3$ .

$54 - 6 y + 3 z - 3 y + 6 z = 18$

$x = 18 - 2 y + z$

$5 x - 8 y + 13 z = 18$

$54 - 6 y + 3 z - 3 y + 6 z = 18$

$x = 18 - 2 y + z$

$5 x - 8 y + 13 z = 18$

$54 - 6 y + 3 z - 3 y + 6 z = 18$

$x = 18 - 2 y + z$

$5 x - 8 y + 13 z = 18$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $3 y$ from $- 6 y$ .

$54 - 9 y + 3 z + 6 z = 18$

$x = 18 - 2 y + z$

$5 x - 8 y + 13 z = 18$

Step 2.2.1.2.2

Add $3 z$ and $6 z$ .

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

$5 x - 8 y + 13 z = 18$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

$5 x - 8 y + 13 z = 18$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

$5 x - 8 y + 13 z = 18$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

$5 x - 8 y + 13 z = 18$

Step 2.3

Replace all occurrences of $x$ in $5 x - 8 y + 13 z = 18$ with $18 - 2 y + z$ .

$5 (18 - 2 y + z) - 8 y + 13 z = 18$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $5 (18 - 2 y + z) - 8 y + 13 z$ .

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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.

$5 \cdot 18 + 5 (- 2 y) + 5 z - 8 y + 13 z = 18$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $5$ by $18$ .

$90 + 5 (- 2 y) + 5 z - 8 y + 13 z = 18$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

Step 2.4.1.1.2.2

Multiply $- 2$ by $5$ .

$90 - 10 y + 5 z - 8 y + 13 z = 18$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

$90 - 10 y + 5 z - 8 y + 13 z = 18$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

$90 - 10 y + 5 z - 8 y + 13 z = 18$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $8 y$ from $- 10 y$ .

$90 - 18 y + 5 z + 13 z = 18$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

Step 2.4.1.2.2

Add $5 z$ and $13 z$ .

$90 - 18 y + 18 z = 18$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

$90 - 18 y + 18 z = 18$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

$90 - 18 y + 18 z = 18$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

$90 - 18 y + 18 z = 18$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

$90 - 18 y + 18 z = 18$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

Step 3

Solve for $y$ in $90 - 18 y + 18 z = 18$ .

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Step 3.1

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

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Step 3.1.1

Subtract $90$ from both sides of the equation.

$- 18 y + 18 z = 18 - 90$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

Step 3.1.2

Subtract $18 z$ from both sides of the equation.

$- 18 y = 18 - 90 - 18 z$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

Step 3.1.3

Subtract $90$ from $18$ .

$- 18 y = - 72 - 18 z$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

$- 18 y = - 72 - 18 z$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

Step 3.2

Divide each term in $- 18 y = - 72 - 18 z$ by $- 18$ and simplify.

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Step 3.2.1

Divide each term in $- 18 y = - 72 - 18 z$ by $- 18$ .

$\frac{- 18 y}{- 18} = \frac{- 72}{- 18} + \frac{- 18 z}{- 18}$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 18$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 18 y}{- 18} = \frac{- 72}{- 18} + \frac{- 18 z}{- 18}$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 72}{- 18} + \frac{- 18 z}{- 18}$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

$y = \frac{- 72}{- 18} + \frac{- 18 z}{- 18}$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

$y = \frac{- 72}{- 18} + \frac{- 18 z}{- 18}$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

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 $- 72$ by $- 18$ .

$y = 4 + \frac{- 18 z}{- 18}$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

Step 3.2.3.1.2

Cancel the common factor of $- 18$ .

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Step 3.2.3.1.2.1

Cancel the common factor.

$y = 4 + \frac{- 18 z}{- 18}$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

Step 3.2.3.1.2.2

Divide $z$ by $1$ .

$y = 4 + z$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

$y = 4 + z$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

$y = 4 + z$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

$y = 4 + z$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

$y = 4 + z$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

$y = 4 + z$

$54 - 9 y + 9 z = 18$

$x = 18 - 2 y + z$

Step 4

Replace all occurrences of $y$ with $4 + z$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $54 - 9 y + 9 z = 18$ with $4 + z$ .

$54 - 9 (4 + z) + 9 z = 18$

$y = 4 + z$

$x = 18 - 2 y + z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $54 - 9 (4 + z) + 9 z$ .

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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.

$54 - 9 \cdot 4 - 9 z + 9 z = 18$

$y = 4 + z$

$x = 18 - 2 y + z$

Step 4.2.1.1.2

Multiply $- 9$ by $4$ .

$54 - 36 - 9 z + 9 z = 18$

$y = 4 + z$

$x = 18 - 2 y + z$

$54 - 36 - 9 z + 9 z = 18$

$y = 4 + z$

$x = 18 - 2 y + z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Combine the opposite terms in $54 - 36 - 9 z + 9 z$ .

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Step 4.2.1.2.1.1

Add $- 9 z$ and $9 z$ .

$54 - 36 + 0 = 18$

$y = 4 + z$

$x = 18 - 2 y + z$

Step 4.2.1.2.1.2

Add $54 - 36$ and $0$ .

$54 - 36 = 18$

$y = 4 + z$

$x = 18 - 2 y + z$

$54 - 36 = 18$

$y = 4 + z$

$x = 18 - 2 y + z$

Step 4.2.1.2.2

Subtract $36$ from $54$ .

$18 = 18$

$y = 4 + z$

$x = 18 - 2 y + z$

$18 = 18$

$y = 4 + z$

$x = 18 - 2 y + z$

$18 = 18$

$y = 4 + z$

$x = 18 - 2 y + z$

$18 = 18$

$y = 4 + z$

$x = 18 - 2 y + z$

Step 4.3

Replace all occurrences of $y$ in $x = 18 - 2 y + z$ with $4 + z$ .

$x = 18 - 2 (4 + z) + z$

$18 = 18$

$y = 4 + z$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $18 - 2 (4 + z) + z$ .

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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.

$x = 18 - 2 \cdot 4 - 2 z + z$

$18 = 18$

$y = 4 + z$

Step 4.4.1.1.2

Multiply $- 2$ by $4$ .

$x = 18 - 8 - 2 z + z$

$18 = 18$

$y = 4 + z$

$x = 18 - 8 - 2 z + z$

$18 = 18$

$y = 4 + z$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Subtract $8$ from $18$ .

$x = 10 - 2 z + z$

$18 = 18$

$y = 4 + z$

Step 4.4.1.2.2

Add $- 2 z$ and $z$ .

$x = 10 - z$

$18 = 18$

$y = 4 + z$

$x = 10 - z$

$18 = 18$

$y = 4 + z$

$x = 10 - z$

$18 = 18$

$y = 4 + z$

$x = 10 - z$

$18 = 18$

$y = 4 + z$

$x = 10 - z$

$18 = 18$

$y = 4 + z$

Step 5

Remove any equations from the system that are always true.

$x = 10 - z$

$y = 4 + z$