Precalculus Examples

$x + y + z = 5$ , $x - y + 3 z = 1$ , $x + 3 y - z = 8$

Step 1

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

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Step 1.1

Subtract $y$ from both sides of the equation.

$x + z = 5 - y$

$x - y + 3 z = 1$

$x + 3 y - z = 8$

Step 1.2

Subtract $z$ from both sides of the equation.

$x = 5 - y - z$

$x - y + 3 z = 1$

$x + 3 y - z = 8$

$x = 5 - y - z$

$x - y + 3 z = 1$

$x + 3 y - z = 8$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $x - y + 3 z = 1$ with $5 - y - z$ .

$(5 - y - z) - y + 3 z = 1$

$x = 5 - y - z$

$x + 3 y - z = 8$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $(5 - y - z) - y + 3 z$ .

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Step 2.2.1.1

Subtract $y$ from $- y$ .

$5 - 2 y - z + 3 z = 1$

$x = 5 - y - z$

$x + 3 y - z = 8$

Step 2.2.1.2

Add $- z$ and $3 z$ .

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

$x + 3 y - z = 8$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

$x + 3 y - z = 8$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

$x + 3 y - z = 8$

Step 2.3

Replace all occurrences of $x$ in $x + 3 y - z = 8$ with $5 - y - z$ .

$(5 - y - z) + 3 y - z = 8$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $(5 - y - z) + 3 y - z$ .

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Step 2.4.1.1

Add $- y$ and $3 y$ .

$5 + 2 y - z - z = 8$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

Step 2.4.1.2

Subtract $z$ from $- z$ .

$5 + 2 y - 2 z = 8$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

$5 + 2 y - 2 z = 8$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

$5 + 2 y - 2 z = 8$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

$5 + 2 y - 2 z = 8$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

Step 3

Solve for $y$ in $5 + 2 y - 2 z = 8$ .

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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 $5$ from both sides of the equation.

$2 y - 2 z = 8 - 5$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

Step 3.1.2

Add $2 z$ to both sides of the equation.

$2 y = 8 - 5 + 2 z$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

Step 3.1.3

Subtract $5$ from $8$ .

$2 y = 3 + 2 z$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

$2 y = 3 + 2 z$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

Step 3.2

Divide each term in $2 y = 3 + 2 z$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 y = 3 + 2 z$ by $2$ .

$\frac{2 y}{2} = \frac{3}{2} + \frac{2 z}{2}$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{3}{2} + \frac{2 z}{2}$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{3}{2} + \frac{2 z}{2}$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

$y = \frac{3}{2} + \frac{2 z}{2}$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

$y = \frac{3}{2} + \frac{2 z}{2}$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $2$ .

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Step 3.2.3.1.1

Cancel the common factor.

$y = \frac{3}{2} + \frac{2 z}{2}$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

Step 3.2.3.1.2

Divide $z$ by $1$ .

$y = \frac{3}{2} + z$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

$y = \frac{3}{2} + z$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

$y = \frac{3}{2} + z$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

$y = \frac{3}{2} + z$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

$y = \frac{3}{2} + z$

$5 - 2 y + 2 z = 1$

$x = 5 - y - z$

Step 4

Replace all occurrences of $y$ with $\frac{3}{2} + z$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $5 - 2 y + 2 z = 1$ with $\frac{3}{2} + z$ .

$5 - 2 (\frac{3}{2} + z) + 2 z = 1$

$y = \frac{3}{2} + z$

$x = 5 - y - z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $5 - 2 (\frac{3}{2} + z) + 2 z$ .

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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.

$5 - 2 (\frac{3}{2}) - 2 z + 2 z = 1$

$y = \frac{3}{2} + z$

$x = 5 - y - z$

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Factor $2$ out of $- 2$ .

$5 + 2 (- 1) (\frac{3}{2}) - 2 z + 2 z = 1$

$y = \frac{3}{2} + z$

$x = 5 - y - z$

Step 4.2.1.1.2.2

Cancel the common factor.

$5 + 2 \cdot (- 1 (\frac{3}{2})) - 2 z + 2 z = 1$

$y = \frac{3}{2} + z$

$x = 5 - y - z$

Step 4.2.1.1.2.3

Rewrite the expression.

$5 - 1 \cdot 3 - 2 z + 2 z = 1$

$y = \frac{3}{2} + z$

$x = 5 - y - z$

$5 - 1 \cdot 3 - 2 z + 2 z = 1$

$y = \frac{3}{2} + z$

$x = 5 - y - z$

Step 4.2.1.1.3

Multiply $- 1$ by $3$ .

$5 - 3 - 2 z + 2 z = 1$

$y = \frac{3}{2} + z$

$x = 5 - y - z$

$5 - 3 - 2 z + 2 z = 1$

$y = \frac{3}{2} + z$

$x = 5 - y - z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Combine the opposite terms in $5 - 3 - 2 z + 2 z$ .

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Step 4.2.1.2.1.1

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

$5 - 3 + 0 = 1$

$y = \frac{3}{2} + z$

$x = 5 - y - z$

Step 4.2.1.2.1.2

Add $5 - 3$ and $0$ .

$5 - 3 = 1$

$y = \frac{3}{2} + z$

$x = 5 - y - z$

$5 - 3 = 1$

$y = \frac{3}{2} + z$

$x = 5 - y - z$

Step 4.2.1.2.2

Subtract $3$ from $5$ .

$2 = 1$

$y = \frac{3}{2} + z$

$x = 5 - y - z$

$2 = 1$

$y = \frac{3}{2} + z$

$x = 5 - y - z$

$2 = 1$

$y = \frac{3}{2} + z$

$x = 5 - y - z$

$2 = 1$

$y = \frac{3}{2} + z$

$x = 5 - y - z$

Step 4.3

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

$x = 5 - (\frac{3}{2} + z) - z$

$2 = 1$

$y = \frac{3}{2} + z$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $5 - (\frac{3}{2} + z) - z$ .

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Step 4.4.1.1

Apply the distributive property.

$x = 5 - \frac{3}{2} - z - z$

$2 = 1$

$y = \frac{3}{2} + z$

Step 4.4.1.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = 5 \cdot \frac{2}{2} - \frac{3}{2} - z - z$

$2 = 1$

$y = \frac{3}{2} + z$

Step 4.4.1.3

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

$x = \frac{5 \cdot 2}{2} - \frac{3}{2} - z - z$

$2 = 1$

$y = \frac{3}{2} + z$

Step 4.4.1.4

Combine the numerators over the common denominator.

$x = \frac{5 \cdot 2 - 3}{2} - z - z$

$2 = 1$

$y = \frac{3}{2} + z$

Step 4.4.1.5

Simplify the numerator.

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Step 4.4.1.5.1

Multiply $5$ by $2$ .

$x = \frac{10 - 3}{2} - z - z$

$2 = 1$

$y = \frac{3}{2} + z$

Step 4.4.1.5.2

Subtract $3$ from $10$ .

$x = \frac{7}{2} - z - z$

$2 = 1$

$y = \frac{3}{2} + z$

$x = \frac{7}{2} - z - z$

$2 = 1$

$y = \frac{3}{2} + z$

Step 4.4.1.6

Subtract $z$ from $- z$ .

$x = \frac{7}{2} - 2 z$

$2 = 1$

$y = \frac{3}{2} + z$

$x = \frac{7}{2} - 2 z$

$2 = 1$

$y = \frac{3}{2} + z$

$x = \frac{7}{2} - 2 z$

$2 = 1$

$y = \frac{3}{2} + z$

$x = \frac{7}{2} - 2 z$

$2 = 1$

$y = \frac{3}{2} + z$

Step 5

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

No solution