Algebra Examples

$4 x - 4 z = 12$ $- 6 x + 4 y = - 28$ $- 2 y + 3 z = 5$

Step 1

Solve for $x$ in $4 x - 4 z = 12$ .

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Step 1.1

Add $4 z$ to both sides of the equation.

$4 x = 12 + 4 z$

$- 6 x + 4 y = - 28$

$- 2 y + 3 z = 5$

Step 1.2

Divide each term in $4 x = 12 + 4 z$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x = 12 + 4 z$ by $4$ .

$\frac{4 x}{4} = \frac{12}{4} + \frac{4 z}{4}$

$- 6 x + 4 y = - 28$

$- 2 y + 3 z = 5$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{12}{4} + \frac{4 z}{4}$

$- 6 x + 4 y = - 28$

$- 2 y + 3 z = 5$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{12}{4} + \frac{4 z}{4}$

$- 6 x + 4 y = - 28$

$- 2 y + 3 z = 5$

$x = \frac{12}{4} + \frac{4 z}{4}$

$- 6 x + 4 y = - 28$

$- 2 y + 3 z = 5$

$x = \frac{12}{4} + \frac{4 z}{4}$

$- 6 x + 4 y = - 28$

$- 2 y + 3 z = 5$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Divide $12$ by $4$ .

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

$- 6 x + 4 y = - 28$

$- 2 y + 3 z = 5$

Step 1.2.3.1.2

Cancel the common factor of $4$ .

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Step 1.2.3.1.2.1

Cancel the common factor.

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

$- 6 x + 4 y = - 28$

$- 2 y + 3 z = 5$

Step 1.2.3.1.2.2

Divide $z$ by $1$ .

$x = 3 + z$

$- 6 x + 4 y = - 28$

$- 2 y + 3 z = 5$

$x = 3 + z$

$- 6 x + 4 y = - 28$

$- 2 y + 3 z = 5$

$x = 3 + z$

$- 6 x + 4 y = - 28$

$- 2 y + 3 z = 5$

$x = 3 + z$

$- 6 x + 4 y = - 28$

$- 2 y + 3 z = 5$

$x = 3 + z$

$- 6 x + 4 y = - 28$

$- 2 y + 3 z = 5$

$x = 3 + z$

$- 6 x + 4 y = - 28$

$- 2 y + 3 z = 5$

Step 2

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

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Step 2.1

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

$- 6 (3 + z) + 4 y = - 28$

$x = 3 + z$

$- 2 y + 3 z = 5$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Apply the distributive property.

$- 6 \cdot 3 - 6 z + 4 y = - 28$

$x = 3 + z$

$- 2 y + 3 z = 5$

Step 2.2.1.2

Multiply $- 6$ by $3$ .

$- 18 - 6 z + 4 y = - 28$

$x = 3 + z$

$- 2 y + 3 z = 5$

$- 18 - 6 z + 4 y = - 28$

$x = 3 + z$

$- 2 y + 3 z = 5$

$- 18 - 6 z + 4 y = - 28$

$x = 3 + z$

$- 2 y + 3 z = 5$

$- 18 - 6 z + 4 y = - 28$

$x = 3 + z$

$- 2 y + 3 z = 5$

Step 3

Reorder $3$ and $z$ .

$x = z + 3$

$- 18 - 6 z + 4 y = - 28$

$- 2 y + 3 z = 5$

Step 4

Solve for $z$ in $x = z + 3$ .

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Step 4.1

Rewrite the equation as $z + 3 = x$ .

$z + 3 = x$

$- 18 - 6 z + 4 y = - 28$

$- 2 y + 3 z = 5$

Step 4.2

Subtract $3$ from both sides of the equation.

$z = x - 3$

$- 18 - 6 z + 4 y = - 28$

$- 2 y + 3 z = 5$

$z = x - 3$

$- 18 - 6 z + 4 y = - 28$

$- 2 y + 3 z = 5$

Step 5

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

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Step 5.1

Replace all occurrences of $z$ in $- 18 - 6 z + 4 y = - 28$ with $x - 3$ .

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

$z = x - 3$

$- 2 y + 3 z = 5$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify $- 18 - 6 (x - 3) + 4 y$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Apply the distributive property.

$- 18 - 6 x - 6 \cdot - 3 + 4 y = - 28$

$z = x - 3$

$- 2 y + 3 z = 5$

Step 5.2.1.1.2

Multiply $- 6$ by $- 3$ .

$- 18 - 6 x + 18 + 4 y = - 28$

$z = x - 3$

$- 2 y + 3 z = 5$

$- 18 - 6 x + 18 + 4 y = - 28$

$z = x - 3$

$- 2 y + 3 z = 5$

Step 5.2.1.2

Combine the opposite terms in $- 18 - 6 x + 18 + 4 y$ .

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Step 5.2.1.2.1

Add $- 18$ and $18$ .

$- 6 x + 0 + 4 y = - 28$

$z = x - 3$

$- 2 y + 3 z = 5$

Step 5.2.1.2.2

Add $- 6 x$ and $0$ .

$- 6 x + 4 y = - 28$

$z = x - 3$

$- 2 y + 3 z = 5$

$- 6 x + 4 y = - 28$

$z = x - 3$

$- 2 y + 3 z = 5$

$- 6 x + 4 y = - 28$

$z = x - 3$

$- 2 y + 3 z = 5$

$- 6 x + 4 y = - 28$

$z = x - 3$

$- 2 y + 3 z = 5$

Step 5.3

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

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

$- 6 x + 4 y = - 28$

$z = x - 3$

Step 5.4

Simplify the left side.

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Step 5.4.1

Simplify each term.

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Step 5.4.1.1

Apply the distributive property.

$- 2 y + 3 x + 3 \cdot - 3 = 5$

$- 6 x + 4 y = - 28$

$z = x - 3$

Step 5.4.1.2

Multiply $3$ by $- 3$ .

$- 2 y + 3 x - 9 = 5$

$- 6 x + 4 y = - 28$

$z = x - 3$

$- 2 y + 3 x - 9 = 5$

$- 6 x + 4 y = - 28$

$z = x - 3$

$- 2 y + 3 x - 9 = 5$

$- 6 x + 4 y = - 28$

$z = x - 3$

$- 2 y + 3 x - 9 = 5$

$- 6 x + 4 y = - 28$

$z = x - 3$

Step 6

Solve for $y$ in $- 2 y + 3 x - 9 = 5$ .

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Step 6.1

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

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Step 6.1.1

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

$- 2 y - 9 = 5 - 3 x$

$- 6 x + 4 y = - 28$

$z = x - 3$

Step 6.1.2

Add $9$ to both sides of the equation.

$- 2 y = 5 - 3 x + 9$

$- 6 x + 4 y = - 28$

$z = x - 3$

Step 6.1.3

Add $5$ and $9$ .

$- 2 y = - 3 x + 14$

$- 6 x + 4 y = - 28$

$z = x - 3$

$- 2 y = - 3 x + 14$

$- 6 x + 4 y = - 28$

$z = x - 3$

Step 6.2

Divide each term in $- 2 y = - 3 x + 14$ by $- 2$ and simplify.

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Step 6.2.1

Divide each term in $- 2 y = - 3 x + 14$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{- 3 x}{- 2} + \frac{14}{- 2}$

$- 6 x + 4 y = - 28$

$z = x - 3$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $- 2$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{- 3 x}{- 2} + \frac{14}{- 2}$

$- 6 x + 4 y = - 28$

$z = x - 3$

Step 6.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 3 x}{- 2} + \frac{14}{- 2}$

$- 6 x + 4 y = - 28$

$z = x - 3$

$y = \frac{- 3 x}{- 2} + \frac{14}{- 2}$

$- 6 x + 4 y = - 28$

$z = x - 3$

$y = \frac{- 3 x}{- 2} + \frac{14}{- 2}$

$- 6 x + 4 y = - 28$

$z = x - 3$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Simplify each term.

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Step 6.2.3.1.1

Dividing two negative values results in a positive value.

$y = \frac{3 x}{2} + \frac{14}{- 2}$

$- 6 x + 4 y = - 28$

$z = x - 3$

Step 6.2.3.1.2

Divide $14$ by $- 2$ .

$y = \frac{3 x}{2} - 7$

$- 6 x + 4 y = - 28$

$z = x - 3$

$y = \frac{3 x}{2} - 7$

$- 6 x + 4 y = - 28$

$z = x - 3$

$y = \frac{3 x}{2} - 7$

$- 6 x + 4 y = - 28$

$z = x - 3$

$y = \frac{3 x}{2} - 7$

$- 6 x + 4 y = - 28$

$z = x - 3$

$y = \frac{3 x}{2} - 7$

$- 6 x + 4 y = - 28$

$z = x - 3$

Step 7

Replace all occurrences of $y$ with $\frac{3 x}{2} - 7$ in each equation.

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Step 7.1

Replace all occurrences of $y$ in $- 6 x + 4 y = - 28$ with $\frac{3 x}{2} - 7$ .

$- 6 x + 4 (\frac{3 x}{2} - 7) = - 28$

$y = \frac{3 x}{2} - 7$

$z = x - 3$

Step 7.2

Simplify the left side.

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Step 7.2.1

Simplify $- 6 x + 4 (\frac{3 x}{2} - 7)$ .

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Step 7.2.1.1

Simplify each term.

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Step 7.2.1.1.1

Apply the distributive property.

$- 6 x + 4 (\frac{3 x}{2}) + 4 \cdot - 7 = - 28$

$y = \frac{3 x}{2} - 7$

$z = x - 3$

Step 7.2.1.1.2

Cancel the common factor of $2$ .

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Step 7.2.1.1.2.1

Factor $2$ out of $4$ .

$- 6 x + 2 (2) (\frac{3 x}{2}) + 4 \cdot - 7 = - 28$

$y = \frac{3 x}{2} - 7$

$z = x - 3$

Step 7.2.1.1.2.2

Cancel the common factor.

$- 6 x + 2 \cdot (2 (\frac{3 x}{2})) + 4 \cdot - 7 = - 28$

$y = \frac{3 x}{2} - 7$

$z = x - 3$

Step 7.2.1.1.2.3

Rewrite the expression.

$- 6 x + 2 (3 x) + 4 \cdot - 7 = - 28$

$y = \frac{3 x}{2} - 7$

$z = x - 3$

$- 6 x + 2 (3 x) + 4 \cdot - 7 = - 28$

$y = \frac{3 x}{2} - 7$

$z = x - 3$

Step 7.2.1.1.3

Multiply $3$ by $2$ .

$- 6 x + 6 x + 4 \cdot - 7 = - 28$

$y = \frac{3 x}{2} - 7$

$z = x - 3$

Step 7.2.1.1.4

Multiply $4$ by $- 7$ .

$- 6 x + 6 x - 28 = - 28$

$y = \frac{3 x}{2} - 7$

$z = x - 3$

$- 6 x + 6 x - 28 = - 28$

$y = \frac{3 x}{2} - 7$

$z = x - 3$

Step 7.2.1.2

Simplify by adding terms.

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Step 7.2.1.2.1

Add $- 6 x$ and $6 x$ .

$0 - 28 = - 28$

$y = \frac{3 x}{2} - 7$

$z = x - 3$

Step 7.2.1.2.2

Subtract $28$ from $0$ .

$- 28 = - 28$

$y = \frac{3 x}{2} - 7$

$z = x - 3$

$- 28 = - 28$

$y = \frac{3 x}{2} - 7$

$z = x - 3$

$- 28 = - 28$

$y = \frac{3 x}{2} - 7$

$z = x - 3$

$- 28 = - 28$

$y = \frac{3 x}{2} - 7$

$z = x - 3$

$- 28 = - 28$

$y = \frac{3 x}{2} - 7$

$z = x - 3$

Step 8

Remove any equations from the system that are always true.

$y = \frac{3 x}{2} - 7$

$z = x - 3$