Finite Math Examples

$2 x + 3 y - z = 2$ , $5 x + 3 z = 7$ , $- 6 x - 9 y + 3 z = 12$

Step 1

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

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Step 1.1

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

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Step 1.1.1

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

$2 x - z = 2 - 3 y$

$5 x + 3 z = 7$

$- 6 x - 9 y + 3 z = 12$

Step 1.1.2

Add $z$ to both sides of the equation.

$2 x = 2 - 3 y + z$

$5 x + 3 z = 7$

$- 6 x - 9 y + 3 z = 12$

$2 x = 2 - 3 y + z$

$5 x + 3 z = 7$

$- 6 x - 9 y + 3 z = 12$

Step 1.2

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

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Step 1.2.1

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

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

$5 x + 3 z = 7$

$- 6 x - 9 y + 3 z = 12$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

$5 x + 3 z = 7$

$- 6 x - 9 y + 3 z = 12$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$5 x + 3 z = 7$

$- 6 x - 9 y + 3 z = 12$

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

$5 x + 3 z = 7$

$- 6 x - 9 y + 3 z = 12$

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

$5 x + 3 z = 7$

$- 6 x - 9 y + 3 z = 12$

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 $2$ by $2$ .

$x = 1 + \frac{- 3 y}{2} + \frac{z}{2}$

$5 x + 3 z = 7$

$- 6 x - 9 y + 3 z = 12$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$5 x + 3 z = 7$

$- 6 x - 9 y + 3 z = 12$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$5 x + 3 z = 7$

$- 6 x - 9 y + 3 z = 12$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$5 x + 3 z = 7$

$- 6 x - 9 y + 3 z = 12$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$5 x + 3 z = 7$

$- 6 x - 9 y + 3 z = 12$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$5 x + 3 z = 7$

$- 6 x - 9 y + 3 z = 12$

Step 2

Replace all occurrences of $x$ with $1 - \frac{3 y}{2} + \frac{z}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $5 x + 3 z = 7$ with $1 - \frac{3 y}{2} + \frac{z}{2}$ .

$5 (1 - \frac{3 y}{2} + \frac{z}{2}) + 3 z = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $5 (1 - \frac{3 y}{2} + \frac{z}{2}) + 3 z$ .

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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.

$5 \cdot 1 + 5 (- \frac{3 y}{2}) + 5 (\frac{z}{2}) + 3 z = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $5$ by $1$ .

$5 + 5 (- \frac{3 y}{2}) + 5 (\frac{z}{2}) + 3 z = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

Step 2.2.1.1.2.2

Multiply $5 (- \frac{3 y}{2})$ .

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Step 2.2.1.1.2.2.1

Multiply $- 1$ by $5$ .

$5 - 5 \frac{3 y}{2} + 5 (\frac{z}{2}) + 3 z = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

Step 2.2.1.1.2.2.2

Combine $- 5$ and $\frac{3 y}{2}$ .

$5 + \frac{- 5 (3 y)}{2} + 5 (\frac{z}{2}) + 3 z = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

Step 2.2.1.1.2.2.3

Multiply $3$ by $- 5$ .

$5 + \frac{- 15 y}{2} + 5 (\frac{z}{2}) + 3 z = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

$5 + \frac{- 15 y}{2} + 5 (\frac{z}{2}) + 3 z = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

Step 2.2.1.1.2.3

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

$5 + \frac{- 15 y}{2} + \frac{5 z}{2} + 3 z = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

$5 + \frac{- 15 y}{2} + \frac{5 z}{2} + 3 z = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

Step 2.2.1.1.3

Move the negative in front of the fraction.

$5 - \frac{15 y}{2} + \frac{5 z}{2} + 3 z = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

$5 - \frac{15 y}{2} + \frac{5 z}{2} + 3 z = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

Step 2.2.1.2

To write $3 z$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$5 - \frac{15 y}{2} + \frac{5 z}{2} + 3 z \cdot \frac{2}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

Step 2.2.1.3

Combine $3 z$ and $\frac{2}{2}$ .

$5 - \frac{15 y}{2} + \frac{5 z}{2} + \frac{3 z \cdot 2}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

Step 2.2.1.4

Combine the numerators over the common denominator.

$5 - \frac{15 y}{2} + \frac{5 z + 3 z \cdot 2}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

Step 2.2.1.5

Combine the numerators over the common denominator.

$5 + \frac{- 15 y + 5 z + 3 z \cdot 2}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

Step 2.2.1.6

Multiply $2$ by $3$ .

$5 + \frac{- 15 y + 5 z + 6 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

Step 2.2.1.7

Add $5 z$ and $6 z$ .

$5 + \frac{- 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

Step 2.2.1.8

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

$5 \cdot \frac{2}{2} + \frac{- 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

Step 2.2.1.9

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

$\frac{5 \cdot 2}{2} + \frac{- 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

Step 2.2.1.10

Combine the numerators over the common denominator.

$\frac{5 \cdot 2 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

Step 2.2.1.11

Multiply $5$ by $2$ .

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 x - 9 y + 3 z = 12$

Step 2.3

Replace all occurrences of $x$ in $- 6 x - 9 y + 3 z = 12$ with $1 - \frac{3 y}{2} + \frac{z}{2}$ .

$- 6 (1 - \frac{3 y}{2} + \frac{z}{2}) - 9 y + 3 z = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $- 6 (1 - \frac{3 y}{2} + \frac{z}{2}) - 9 y + 3 z$ .

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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.

$- 6 \cdot 1 - 6 (- \frac{3 y}{2}) - 6 \frac{z}{2} - 9 y + 3 z = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $- 6$ by $1$ .

$- 6 - 6 (- \frac{3 y}{2}) - 6 \frac{z}{2} - 9 y + 3 z = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

Step 2.4.1.1.2.2

Cancel the common factor of $2$ .

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Step 2.4.1.1.2.2.1

Move the leading negative in $- \frac{3 y}{2}$ into the numerator.

$- 6 - 6 \frac{- 3 y}{2} - 6 \frac{z}{2} - 9 y + 3 z = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

Step 2.4.1.1.2.2.2

Factor $2$ out of $- 6$ .

$- 6 + 2 (- 3) (\frac{- 3 y}{2}) - 6 \frac{z}{2} - 9 y + 3 z = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

Step 2.4.1.1.2.2.3

Cancel the common factor.

$- 6 + 2 \cdot (- 3 \frac{- 3 y}{2}) - 6 \frac{z}{2} - 9 y + 3 z = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

Step 2.4.1.1.2.2.4

Rewrite the expression.

$- 6 - 3 (- 3 y) - 6 \frac{z}{2} - 9 y + 3 z = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 - 3 (- 3 y) - 6 \frac{z}{2} - 9 y + 3 z = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

Step 2.4.1.1.2.3

Multiply $- 3$ by $- 3$ .

$- 6 + 9 y - 6 \frac{z}{2} - 9 y + 3 z = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

Step 2.4.1.1.2.4

Cancel the common factor of $2$ .

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Step 2.4.1.1.2.4.1

Factor $2$ out of $- 6$ .

$- 6 + 9 y + 2 (- 3) (\frac{z}{2}) - 9 y + 3 z = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

Step 2.4.1.1.2.4.2

Cancel the common factor.

$- 6 + 9 y + 2 \cdot (- 3 \frac{z}{2}) - 9 y + 3 z = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

Step 2.4.1.1.2.4.3

Rewrite the expression.

$- 6 + 9 y - 3 z - 9 y + 3 z = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 + 9 y - 3 z - 9 y + 3 z = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 + 9 y - 3 z - 9 y + 3 z = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 + 9 y - 3 z - 9 y + 3 z = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

Step 2.4.1.2

Combine the opposite terms in $- 6 + 9 y - 3 z - 9 y + 3 z$ .

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Step 2.4.1.2.1

Subtract $9 y$ from $9 y$ .

$- 6 - 3 z + 0 + 3 z = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

Step 2.4.1.2.2

Add $- 6 - 3 z$ and $0$ .

$- 6 - 3 z + 3 z = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

Step 2.4.1.2.3

Add $- 3 z$ and $3 z$ .

$- 6 + 0 = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

Step 2.4.1.2.4

Add $- 6$ and $0$ .

$- 6 = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

$- 6 = 12$

$\frac{10 - 15 y + 11 z}{2} = 7$

$x = 1 - \frac{3 y}{2} + \frac{z}{2}$

Step 3

Since $- 6 = 12$ is not true, there is no solution.

No solution