Finite Math Examples

$x + y - z = 7$ , $3 y + 2 z = 1$ , $- 2 x - 5 y = 6$

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.

$x - z = 7 - y$

$3 y + 2 z = 1$

$- 2 x - 5 y = 6$

Step 1.2

Add $z$ to both sides of the equation.

$x = 7 - y + z$

$3 y + 2 z = 1$

$- 2 x - 5 y = 6$

$x = 7 - y + z$

$3 y + 2 z = 1$

$- 2 x - 5 y = 6$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $- 2 x - 5 y = 6$ with $7 - y + z$ .

$- 2 (7 - y + z) - 5 y = 6$

$x = 7 - y + z$

$3 y + 2 z = 1$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 2 (7 - y + z) - 5 y$ .

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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.

$- 2 \cdot 7 - 2 (- y) - 2 z - 5 y = 6$

$x = 7 - y + z$

$3 y + 2 z = 1$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 2$ by $7$ .

$- 14 - 2 (- y) - 2 z - 5 y = 6$

$x = 7 - y + z$

$3 y + 2 z = 1$

Step 2.2.1.1.2.2

Multiply $- 1$ by $- 2$ .

$- 14 + 2 y - 2 z - 5 y = 6$

$x = 7 - y + z$

$3 y + 2 z = 1$

$- 14 + 2 y - 2 z - 5 y = 6$

$x = 7 - y + z$

$3 y + 2 z = 1$

$- 14 + 2 y - 2 z - 5 y = 6$

$x = 7 - y + z$

$3 y + 2 z = 1$

Step 2.2.1.2

Subtract $5 y$ from $2 y$ .

$- 14 - 3 y - 2 z = 6$

$x = 7 - y + z$

$3 y + 2 z = 1$

$- 14 - 3 y - 2 z = 6$

$x = 7 - y + z$

$3 y + 2 z = 1$

$- 14 - 3 y - 2 z = 6$

$x = 7 - y + z$

$3 y + 2 z = 1$

$- 14 - 3 y - 2 z = 6$

$x = 7 - y + z$

$3 y + 2 z = 1$

Step 3

Solve for $y$ in $- 14 - 3 y - 2 z = 6$ .

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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

Add $14$ to both sides of the equation.

$- 3 y - 2 z = 6 + 14$

$x = 7 - y + z$

$3 y + 2 z = 1$

Step 3.1.2

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

$- 3 y = 6 + 14 + 2 z$

$x = 7 - y + z$

$3 y + 2 z = 1$

Step 3.1.3

Add $6$ and $14$ .

$- 3 y = 20 + 2 z$

$x = 7 - y + z$

$3 y + 2 z = 1$

$- 3 y = 20 + 2 z$

$x = 7 - y + z$

$3 y + 2 z = 1$

Step 3.2

Divide each term in $- 3 y = 20 + 2 z$ by $- 3$ and simplify.

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Step 3.2.1

Divide each term in $- 3 y = 20 + 2 z$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{20}{- 3} + \frac{2 z}{- 3}$

$x = 7 - y + z$

$3 y + 2 z = 1$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 3$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{20}{- 3} + \frac{2 z}{- 3}$

$x = 7 - y + z$

$3 y + 2 z = 1$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{20}{- 3} + \frac{2 z}{- 3}$

$x = 7 - y + z$

$3 y + 2 z = 1$

$y = \frac{20}{- 3} + \frac{2 z}{- 3}$

$x = 7 - y + z$

$3 y + 2 z = 1$

$y = \frac{20}{- 3} + \frac{2 z}{- 3}$

$x = 7 - y + z$

$3 y + 2 z = 1$

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

Move the negative in front of the fraction.

$y = - \frac{20}{3} + \frac{2 z}{- 3}$

$x = 7 - y + z$

$3 y + 2 z = 1$

Step 3.2.3.1.2

Move the negative in front of the fraction.

$y = - \frac{20}{3} - \frac{2 z}{3}$

$x = 7 - y + z$

$3 y + 2 z = 1$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$x = 7 - y + z$

$3 y + 2 z = 1$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$x = 7 - y + z$

$3 y + 2 z = 1$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$x = 7 - y + z$

$3 y + 2 z = 1$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$x = 7 - y + z$

$3 y + 2 z = 1$

Step 4

Replace all occurrences of $y$ with $- \frac{20}{3} - \frac{2 z}{3}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 7 - y + z$ with $- \frac{20}{3} - \frac{2 z}{3}$ .

$x = 7 - (- \frac{20}{3} - \frac{2 z}{3}) + z$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $7 - (- \frac{20}{3} - \frac{2 z}{3}) + z$ .

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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.

$x = 7 + \frac{20}{3} + \frac{2 z}{3} + z$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

Step 4.2.1.1.2

Multiply $- - \frac{20}{3}$ .

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Step 4.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$x = 7 + 1 (\frac{20}{3}) + \frac{2 z}{3} + z$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

Step 4.2.1.1.2.2

Multiply $\frac{20}{3}$ by $1$ .

$x = 7 + \frac{20}{3} + \frac{2 z}{3} + z$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

$x = 7 + \frac{20}{3} + \frac{2 z}{3} + z$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

Step 4.2.1.1.3

Multiply $- - \frac{2 z}{3}$ .

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Step 4.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$x = 7 + \frac{20}{3} + 1 (\frac{2 z}{3}) + z$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

Step 4.2.1.1.3.2

Multiply $\frac{2 z}{3}$ by $1$ .

$x = 7 + \frac{20}{3} + \frac{2 z}{3} + z$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

$x = 7 + \frac{20}{3} + \frac{2 z}{3} + z$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

$x = 7 + \frac{20}{3} + \frac{2 z}{3} + z$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

Step 4.2.1.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = 7 \cdot \frac{3}{3} + \frac{20}{3} + \frac{2 z}{3} + z$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

Step 4.2.1.3

Combine $7$ and $\frac{3}{3}$ .

$x = \frac{7 \cdot 3}{3} + \frac{20}{3} + \frac{2 z}{3} + z$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x = \frac{7 \cdot 3 + 20}{3} + \frac{2 z}{3} + z$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $7$ by $3$ .

$x = \frac{21 + 20}{3} + \frac{2 z}{3} + z$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

Step 4.2.1.5.2

Add $21$ and $20$ .

$x = \frac{41}{3} + \frac{2 z}{3} + z$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

$x = \frac{41}{3} + \frac{2 z}{3} + z$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

Step 4.2.1.6

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

$x = \frac{41}{3} + \frac{2 z}{3} + z \cdot \frac{3}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

Step 4.2.1.7

Simplify terms.

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Step 4.2.1.7.1

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

$x = \frac{41}{3} + \frac{2 z}{3} + \frac{z \cdot 3}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

Step 4.2.1.7.2

Combine the numerators over the common denominator.

$x = \frac{41}{3} + \frac{2 z + z \cdot 3}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

Step 4.2.1.7.3

Combine the numerators over the common denominator.

$x = \frac{41 + 2 z + z \cdot 3}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

$x = \frac{41 + 2 z + z \cdot 3}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

Step 4.2.1.8

Move $3$ to the left of $z$ .

$x = \frac{41 + 2 z + 3 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

Step 4.2.1.9

Add $2 z$ and $3 z$ .

$x = \frac{41 + 5 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

$x = \frac{41 + 5 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

$x = \frac{41 + 5 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$3 y + 2 z = 1$

Step 4.3

Replace all occurrences of $y$ in $3 y + 2 z = 1$ with $- \frac{20}{3} - \frac{2 z}{3}$ .

$3 (- \frac{20}{3} - \frac{2 z}{3}) + 2 z = 1$

$x = \frac{41 + 5 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

Step 4.4

Simplify the left side.

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Step 4.4.1

Simplify $3 (- \frac{20}{3} - \frac{2 z}{3}) + 2 z$ .

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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.

$3 (- \frac{20}{3}) + 3 (- \frac{2 z}{3}) + 2 z = 1$

$x = \frac{41 + 5 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

Step 4.4.1.1.2

Cancel the common factor of $3$ .

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Step 4.4.1.1.2.1

Move the leading negative in $- \frac{20}{3}$ into the numerator.

$3 (\frac{- 20}{3}) + 3 (- \frac{2 z}{3}) + 2 z = 1$

$x = \frac{41 + 5 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

Step 4.4.1.1.2.2

Cancel the common factor.

$3 (\frac{- 20}{3}) + 3 (- \frac{2 z}{3}) + 2 z = 1$

$x = \frac{41 + 5 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

Step 4.4.1.1.2.3

Rewrite the expression.

$- 20 + 3 (- \frac{2 z}{3}) + 2 z = 1$

$x = \frac{41 + 5 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$- 20 + 3 (- \frac{2 z}{3}) + 2 z = 1$

$x = \frac{41 + 5 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

Step 4.4.1.1.3

Cancel the common factor of $3$ .

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Step 4.4.1.1.3.1

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

$- 20 + 3 (\frac{- 2 z}{3}) + 2 z = 1$

$x = \frac{41 + 5 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

Step 4.4.1.1.3.2

Cancel the common factor.

$- 20 + 3 (\frac{- 2 z}{3}) + 2 z = 1$

$x = \frac{41 + 5 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

Step 4.4.1.1.3.3

Rewrite the expression.

$- 20 - 2 z + 2 z = 1$

$x = \frac{41 + 5 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$- 20 - 2 z + 2 z = 1$

$x = \frac{41 + 5 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$- 20 - 2 z + 2 z = 1$

$x = \frac{41 + 5 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

Step 4.4.1.2

Combine the opposite terms in $- 20 - 2 z + 2 z$ .

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Step 4.4.1.2.1

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

$- 20 + 0 = 1$

$x = \frac{41 + 5 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

Step 4.4.1.2.2

Add $- 20$ and $0$ .

$- 20 = 1$

$x = \frac{41 + 5 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$- 20 = 1$

$x = \frac{41 + 5 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$- 20 = 1$

$x = \frac{41 + 5 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$- 20 = 1$

$x = \frac{41 + 5 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

$- 20 = 1$

$x = \frac{41 + 5 z}{3}$

$y = - \frac{20}{3} - \frac{2 z}{3}$

Step 5

Since $- 20 = 1$ is not true, there is no solution.

No solution