Algebra Examples

$2 x - 3 y + z = 4$ $y - 2 z + x - 5 = 0$ $3 - 2 x = 4 y - z$

Step 1

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

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Step 1.1

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

$- 3 y + z = 4 - 2 x$

$y - 2 z + x - 5 = 0$

$3 - 2 x = 4 y - z$

Step 1.2

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

$z = 4 - 2 x + 3 y$

$y - 2 z + x - 5 = 0$

$3 - 2 x = 4 y - z$

$z = 4 - 2 x + 3 y$

$y - 2 z + x - 5 = 0$

$3 - 2 x = 4 y - z$

Step 2

Replace all occurrences of $z$ with $4 - 2 x + 3 y$ in each equation.

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Step 2.1

Replace all occurrences of $z$ in $y - 2 z + x - 5 = 0$ with $4 - 2 x + 3 y$ .

$y - 2 (4 - 2 x + 3 y) + x - 5 = 0$

$z = 4 - 2 x + 3 y$

$3 - 2 x = 4 y - z$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $y - 2 (4 - 2 x + 3 y) + x - 5$ .

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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.

$y - 2 \cdot 4 - 2 (- 2 x) - 2 (3 y) + x - 5 = 0$

$z = 4 - 2 x + 3 y$

$3 - 2 x = 4 y - z$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 2$ by $4$ .

$y - 8 - 2 (- 2 x) - 2 (3 y) + x - 5 = 0$

$z = 4 - 2 x + 3 y$

$3 - 2 x = 4 y - z$

Step 2.2.1.1.2.2

Multiply $- 2$ by $- 2$ .

$y - 8 + 4 x - 2 (3 y) + x - 5 = 0$

$z = 4 - 2 x + 3 y$

$3 - 2 x = 4 y - z$

Step 2.2.1.1.2.3

Multiply $3$ by $- 2$ .

$y - 8 + 4 x - 6 y + x - 5 = 0$

$z = 4 - 2 x + 3 y$

$3 - 2 x = 4 y - z$

$y - 8 + 4 x - 6 y + x - 5 = 0$

$z = 4 - 2 x + 3 y$

$3 - 2 x = 4 y - z$

$y - 8 + 4 x - 6 y + x - 5 = 0$

$z = 4 - 2 x + 3 y$

$3 - 2 x = 4 y - z$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Subtract $6 y$ from $y$ .

$- 5 y - 8 + 4 x + x - 5 = 0$

$z = 4 - 2 x + 3 y$

$3 - 2 x = 4 y - z$

Step 2.2.1.2.2

Subtract $5$ from $- 8$ .

$- 5 y + 4 x + x - 13 = 0$

$z = 4 - 2 x + 3 y$

$3 - 2 x = 4 y - z$

Step 2.2.1.2.3

Add $4 x$ and $x$ .

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

$3 - 2 x = 4 y - z$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

$3 - 2 x = 4 y - z$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

$3 - 2 x = 4 y - z$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

$3 - 2 x = 4 y - z$

Step 2.3

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

$3 - 2 x = 4 y - (4 - 2 x + 3 y)$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

Step 2.4

Simplify the right side.

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Step 2.4.1

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

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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.

$3 - 2 x = 4 y - 1 \cdot 4 - (- 2 x) - (3 y)$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 1$ by $4$ .

$3 - 2 x = 4 y - 4 - (- 2 x) - (3 y)$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

Step 2.4.1.1.2.2

Multiply $- 2$ by $- 1$ .

$3 - 2 x = 4 y - 4 + 2 x - (3 y)$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

Step 2.4.1.1.2.3

Multiply $3$ by $- 1$ .

$3 - 2 x = 4 y - 4 + 2 x - 3 y$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

$3 - 2 x = 4 y - 4 + 2 x - 3 y$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

$3 - 2 x = 4 y - 4 + 2 x - 3 y$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

Step 2.4.1.2

Subtract $3 y$ from $4 y$ .

$3 - 2 x = y - 4 + 2 x$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

$3 - 2 x = y - 4 + 2 x$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

$3 - 2 x = y - 4 + 2 x$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

$3 - 2 x = y - 4 + 2 x$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

Step 3

Solve for $y$ in $3 - 2 x = y - 4 + 2 x$ .

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Step 3.1

Rewrite the equation as $y - 4 + 2 x = 3 - 2 x$ .

$y - 4 + 2 x = 3 - 2 x$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

Step 3.2

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

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Step 3.2.1

Add $4$ to both sides of the equation.

$y + 2 x = 3 - 2 x + 4$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

Step 3.2.2

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

$y = 3 - 2 x + 4 - 2 x$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

Step 3.2.3

Add $3$ and $4$ .

$y = - 2 x + 7 - 2 x$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

Step 3.2.4

Subtract $2 x$ from $- 2 x$ .

$y = - 4 x + 7$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

$y = - 4 x + 7$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

$y = - 4 x + 7$

$- 5 y + 5 x - 13 = 0$

$z = 4 - 2 x + 3 y$

Step 4

Replace all occurrences of $y$ with $- 4 x + 7$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $- 5 y + 5 x - 13 = 0$ with $- 4 x + 7$ .

$- 5 (- 4 x + 7) + 5 x - 13 = 0$

$y = - 4 x + 7$

$z = 4 - 2 x + 3 y$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- 5 (- 4 x + 7) + 5 x - 13$ .

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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.

$- 5 (- 4 x) - 5 \cdot 7 + 5 x - 13 = 0$

$y = - 4 x + 7$

$z = 4 - 2 x + 3 y$

Step 4.2.1.1.2

Multiply $- 4$ by $- 5$ .

$20 x - 5 \cdot 7 + 5 x - 13 = 0$

$y = - 4 x + 7$

$z = 4 - 2 x + 3 y$

Step 4.2.1.1.3

Multiply $- 5$ by $7$ .

$20 x - 35 + 5 x - 13 = 0$

$y = - 4 x + 7$

$z = 4 - 2 x + 3 y$

$20 x - 35 + 5 x - 13 = 0$

$y = - 4 x + 7$

$z = 4 - 2 x + 3 y$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Add $20 x$ and $5 x$ .

$25 x - 35 - 13 = 0$

$y = - 4 x + 7$

$z = 4 - 2 x + 3 y$

Step 4.2.1.2.2

Subtract $13$ from $- 35$ .

$25 x - 48 = 0$

$y = - 4 x + 7$

$z = 4 - 2 x + 3 y$

$25 x - 48 = 0$

$y = - 4 x + 7$

$z = 4 - 2 x + 3 y$

$25 x - 48 = 0$

$y = - 4 x + 7$

$z = 4 - 2 x + 3 y$

$25 x - 48 = 0$

$y = - 4 x + 7$

$z = 4 - 2 x + 3 y$

Step 4.3

Replace all occurrences of $y$ in $z = 4 - 2 x + 3 y$ with $- 4 x + 7$ .

$z = 4 - 2 x + 3 (- 4 x + 7)$

$25 x - 48 = 0$

$y = - 4 x + 7$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $4 - 2 x + 3 (- 4 x + 7)$ .

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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.

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

$25 x - 48 = 0$

$y = - 4 x + 7$

Step 4.4.1.1.2

Multiply $- 4$ by $3$ .

$z = 4 - 2 x - 12 x + 3 \cdot 7$

$25 x - 48 = 0$

$y = - 4 x + 7$

Step 4.4.1.1.3

Multiply $3$ by $7$ .

$z = 4 - 2 x - 12 x + 21$

$25 x - 48 = 0$

$y = - 4 x + 7$

$z = 4 - 2 x - 12 x + 21$

$25 x - 48 = 0$

$y = - 4 x + 7$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Add $4$ and $21$ .

$z = - 2 x - 12 x + 25$

$25 x - 48 = 0$

$y = - 4 x + 7$

Step 4.4.1.2.2

Subtract $12 x$ from $- 2 x$ .

$z = - 14 x + 25$

$25 x - 48 = 0$

$y = - 4 x + 7$

$z = - 14 x + 25$

$25 x - 48 = 0$

$y = - 4 x + 7$

$z = - 14 x + 25$

$25 x - 48 = 0$

$y = - 4 x + 7$

$z = - 14 x + 25$

$25 x - 48 = 0$

$y = - 4 x + 7$

$z = - 14 x + 25$

$25 x - 48 = 0$

$y = - 4 x + 7$

Step 5

Solve for $x$ in $25 x - 48 = 0$ .

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Step 5.1

Add $48$ to both sides of the equation.

$25 x = 48$

$z = - 14 x + 25$

$y = - 4 x + 7$

Step 5.2

Divide each term in $25 x = 48$ by $25$ and simplify.

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Step 5.2.1

Divide each term in $25 x = 48$ by $25$ .

$\frac{25 x}{25} = \frac{48}{25}$

$z = - 14 x + 25$

$y = - 4 x + 7$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $25$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{25 x}{25} = \frac{48}{25}$

$z = - 14 x + 25$

$y = - 4 x + 7$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{48}{25}$

$z = - 14 x + 25$

$y = - 4 x + 7$

$x = \frac{48}{25}$

$z = - 14 x + 25$

$y = - 4 x + 7$

$x = \frac{48}{25}$

$z = - 14 x + 25$

$y = - 4 x + 7$

$x = \frac{48}{25}$

$z = - 14 x + 25$

$y = - 4 x + 7$

$x = \frac{48}{25}$

$z = - 14 x + 25$

$y = - 4 x + 7$

Step 6

Replace all occurrences of $x$ with $\frac{48}{25}$ in each equation.

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Step 6.1

Replace all occurrences of $x$ in $z = - 14 x + 25$ with $\frac{48}{25}$ .

$z = - 14 (\frac{48}{25}) + 25$

$x = \frac{48}{25}$

$y = - 4 x + 7$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- 14 (\frac{48}{25}) + 25$ .

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Step 6.2.1.1

Simplify each term.

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Step 6.2.1.1.1

Multiply $- 14 (\frac{48}{25})$ .

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Step 6.2.1.1.1.1

Combine $- 14$ and $\frac{48}{25}$ .

$z = \frac{- 14 \cdot 48}{25} + 25$

$x = \frac{48}{25}$

$y = - 4 x + 7$

Step 6.2.1.1.1.2

Multiply $- 14$ by $48$ .

$z = \frac{- 672}{25} + 25$

$x = \frac{48}{25}$

$y = - 4 x + 7$

$z = \frac{- 672}{25} + 25$

$x = \frac{48}{25}$

$y = - 4 x + 7$

Step 6.2.1.1.2

Move the negative in front of the fraction.

$z = - \frac{672}{25} + 25$

$x = \frac{48}{25}$

$y = - 4 x + 7$

$z = - \frac{672}{25} + 25$

$x = \frac{48}{25}$

$y = - 4 x + 7$

Step 6.2.1.2

To write $25$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$z = - \frac{672}{25} + 25 \cdot \frac{25}{25}$

$x = \frac{48}{25}$

$y = - 4 x + 7$

Step 6.2.1.3

Combine $25$ and $\frac{25}{25}$ .

$z = - \frac{672}{25} + \frac{25 \cdot 25}{25}$

$x = \frac{48}{25}$

$y = - 4 x + 7$

Step 6.2.1.4

Combine the numerators over the common denominator.

$z = \frac{- 672 + 25 \cdot 25}{25}$

$x = \frac{48}{25}$

$y = - 4 x + 7$

Step 6.2.1.5

Simplify the numerator.

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Step 6.2.1.5.1

Multiply $25$ by $25$ .

$z = \frac{- 672 + 625}{25}$

$x = \frac{48}{25}$

$y = - 4 x + 7$

Step 6.2.1.5.2

Add $- 672$ and $625$ .

$z = \frac{- 47}{25}$

$x = \frac{48}{25}$

$y = - 4 x + 7$

$z = \frac{- 47}{25}$

$x = \frac{48}{25}$

$y = - 4 x + 7$

Step 6.2.1.6

Move the negative in front of the fraction.

$z = - \frac{47}{25}$

$x = \frac{48}{25}$

$y = - 4 x + 7$

$z = - \frac{47}{25}$

$x = \frac{48}{25}$

$y = - 4 x + 7$

$z = - \frac{47}{25}$

$x = \frac{48}{25}$

$y = - 4 x + 7$

Step 6.3

Replace all occurrences of $x$ in $y = - 4 x + 7$ with $\frac{48}{25}$ .

$y = - 4 (\frac{48}{25}) + 7$

$z = - \frac{47}{25}$

$x = \frac{48}{25}$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- 4 (\frac{48}{25}) + 7$ .

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Step 6.4.1.1

Simplify each term.

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Step 6.4.1.1.1

Multiply $- 4 (\frac{48}{25})$ .

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Step 6.4.1.1.1.1

Combine $- 4$ and $\frac{48}{25}$ .

$y = \frac{- 4 \cdot 48}{25} + 7$

$z = - \frac{47}{25}$

$x = \frac{48}{25}$

Step 6.4.1.1.1.2

Multiply $- 4$ by $48$ .

$y = \frac{- 192}{25} + 7$

$z = - \frac{47}{25}$

$x = \frac{48}{25}$

$y = \frac{- 192}{25} + 7$

$z = - \frac{47}{25}$

$x = \frac{48}{25}$

Step 6.4.1.1.2

Move the negative in front of the fraction.

$y = - \frac{192}{25} + 7$

$z = - \frac{47}{25}$

$x = \frac{48}{25}$

$y = - \frac{192}{25} + 7$

$z = - \frac{47}{25}$

$x = \frac{48}{25}$

Step 6.4.1.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$y = - \frac{192}{25} + 7 \cdot \frac{25}{25}$

$z = - \frac{47}{25}$

$x = \frac{48}{25}$

Step 6.4.1.3

Combine $7$ and $\frac{25}{25}$ .

$y = - \frac{192}{25} + \frac{7 \cdot 25}{25}$

$z = - \frac{47}{25}$

$x = \frac{48}{25}$

Step 6.4.1.4

Combine the numerators over the common denominator.

$y = \frac{- 192 + 7 \cdot 25}{25}$

$z = - \frac{47}{25}$

$x = \frac{48}{25}$

Step 6.4.1.5

Simplify the numerator.

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Step 6.4.1.5.1

Multiply $7$ by $25$ .

$y = \frac{- 192 + 175}{25}$

$z = - \frac{47}{25}$

$x = \frac{48}{25}$

Step 6.4.1.5.2

Add $- 192$ and $175$ .

$y = \frac{- 17}{25}$

$z = - \frac{47}{25}$

$x = \frac{48}{25}$

$y = \frac{- 17}{25}$

$z = - \frac{47}{25}$

$x = \frac{48}{25}$

Step 6.4.1.6

Move the negative in front of the fraction.

$y = - \frac{17}{25}$

$z = - \frac{47}{25}$

$x = \frac{48}{25}$

$y = - \frac{17}{25}$

$z = - \frac{47}{25}$

$x = \frac{48}{25}$

$y = - \frac{17}{25}$

$z = - \frac{47}{25}$

$x = \frac{48}{25}$

$y = - \frac{17}{25}$

$z = - \frac{47}{25}$

$x = \frac{48}{25}$

Step 7

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

$(\frac{48}{25}, - \frac{17}{25}, - \frac{47}{25})$

Step 8

The result can be shown in multiple forms.

Point Form:

$(\frac{48}{25}, - \frac{17}{25}, - \frac{47}{25})$

Equation Form:

$x = \frac{48}{25}, y = - \frac{17}{25}, z = - \frac{47}{25}$