Finite Math Examples

$2 a - 2 y = 2$ , $8 y + 16 Z = 53$ , $a + 2 Z = 6$

Step 1

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

$a = 6 - 2 Z$

$2 a - 2 y = 2$

$8 y + 16 Z = 53$

Step 2

Replace all occurrences of $a$ with $6 - 2 Z$ in each equation.

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Step 2.1

Replace all occurrences of $a$ in $2 a - 2 y = 2$ with $6 - 2 Z$ .

$2 (6 - 2 Z) - 2 y = 2$

$a = 6 - 2 Z$

$8 y + 16 Z = 53$

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.

$2 \cdot 6 + 2 (- 2 Z) - 2 y = 2$

$a = 6 - 2 Z$

$8 y + 16 Z = 53$

Step 2.2.1.2

Multiply $2$ by $6$ .

$12 + 2 (- 2 Z) - 2 y = 2$

$a = 6 - 2 Z$

$8 y + 16 Z = 53$

Step 2.2.1.3

Multiply $- 2$ by $2$ .

$12 - 4 Z - 2 y = 2$

$a = 6 - 2 Z$

$8 y + 16 Z = 53$

$12 - 4 Z - 2 y = 2$

$a = 6 - 2 Z$

$8 y + 16 Z = 53$

$12 - 4 Z - 2 y = 2$

$a = 6 - 2 Z$

$8 y + 16 Z = 53$

$12 - 4 Z - 2 y = 2$

$a = 6 - 2 Z$

$8 y + 16 Z = 53$

Step 3

Reorder $6$ and $- 2 Z$ .

$a = - 2 Z + 6$

$12 - 4 Z - 2 y = 2$

$8 y + 16 Z = 53$

Step 4

Solve for $Z$ in $12 - 4 Z - 2 y = 2$ .

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Step 4.1

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

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Step 4.1.1

Subtract $12$ from both sides of the equation.

$- 4 Z - 2 y = 2 - 12$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

Step 4.1.2

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

$- 4 Z = 2 - 12 + 2 y$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

Step 4.1.3

Subtract $12$ from $2$ .

$- 4 Z = - 10 + 2 y$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

$- 4 Z = - 10 + 2 y$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

Step 4.2

Divide each term in $- 4 Z = - 10 + 2 y$ by $- 4$ and simplify.

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Step 4.2.1

Divide each term in $- 4 Z = - 10 + 2 y$ by $- 4$ .

$\frac{- 4 Z}{- 4} = \frac{- 10}{- 4} + \frac{2 y}{- 4}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 4$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 4 Z}{- 4} = \frac{- 10}{- 4} + \frac{2 y}{- 4}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

Step 4.2.2.1.2

Divide $Z$ by $1$ .

$Z = \frac{- 10}{- 4} + \frac{2 y}{- 4}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

$Z = \frac{- 10}{- 4} + \frac{2 y}{- 4}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

$Z = \frac{- 10}{- 4} + \frac{2 y}{- 4}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Cancel the common factor of $- 10$ and $- 4$ .

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Step 4.2.3.1.1.1

Factor $- 2$ out of $- 10$ .

$Z = \frac{- 2 \cdot 5}{- 4} + \frac{2 y}{- 4}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

Step 4.2.3.1.1.2

Cancel the common factors.

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Step 4.2.3.1.1.2.1

Factor $- 2$ out of $- 4$ .

$Z = \frac{- 2 \cdot 5}{- 2 \cdot 2} + \frac{2 y}{- 4}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

Step 4.2.3.1.1.2.2

Cancel the common factor.

$Z = \frac{- 2 \cdot 5}{- 2 \cdot 2} + \frac{2 y}{- 4}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

Step 4.2.3.1.1.2.3

Rewrite the expression.

$Z = \frac{5}{2} + \frac{2 y}{- 4}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

$Z = \frac{5}{2} + \frac{2 y}{- 4}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

$Z = \frac{5}{2} + \frac{2 y}{- 4}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

Step 4.2.3.1.2

Cancel the common factor of $2$ and $- 4$ .

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Step 4.2.3.1.2.1

Factor $2$ out of $2 y$ .

$Z = \frac{5}{2} + \frac{2 (y)}{- 4}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

Step 4.2.3.1.2.2

Cancel the common factors.

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Step 4.2.3.1.2.2.1

Factor $2$ out of $- 4$ .

$Z = \frac{5}{2} + \frac{2 y}{2 \cdot - 2}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

Step 4.2.3.1.2.2.2

Cancel the common factor.

$Z = \frac{5}{2} + \frac{2 y}{2 \cdot - 2}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

Step 4.2.3.1.2.2.3

Rewrite the expression.

$Z = \frac{5}{2} + \frac{y}{- 2}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

$Z = \frac{5}{2} + \frac{y}{- 2}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

$Z = \frac{5}{2} + \frac{y}{- 2}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

Step 4.2.3.1.3

Move the negative in front of the fraction.

$Z = \frac{5}{2} - \frac{y}{2}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

$Z = \frac{5}{2} - \frac{y}{2}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

$Z = \frac{5}{2} - \frac{y}{2}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

$Z = \frac{5}{2} - \frac{y}{2}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

$Z = \frac{5}{2} - \frac{y}{2}$

$a = - 2 Z + 6$

$8 y + 16 Z = 53$

Step 5

Replace all occurrences of $Z$ with $\frac{5}{2} - \frac{y}{2}$ in each equation.

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Step 5.1

Replace all occurrences of $Z$ in $a = - 2 Z + 6$ with $\frac{5}{2} - \frac{y}{2}$ .

$a = - 2 (\frac{5}{2} - \frac{y}{2}) + 6$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 16 Z = 53$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 2 (\frac{5}{2} - \frac{y}{2}) + 6$ .

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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.

$a = - 2 (\frac{5}{2}) - 2 (- \frac{y}{2}) + 6$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 16 Z = 53$

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

Factor $2$ out of $- 2$ .

$a = 2 (- 1) (\frac{5}{2}) - 2 (- \frac{y}{2}) + 6$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 16 Z = 53$

Step 5.2.1.1.2.2

Cancel the common factor.

$a = 2 \cdot (- 1 (\frac{5}{2})) - 2 (- \frac{y}{2}) + 6$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 16 Z = 53$

Step 5.2.1.1.2.3

Rewrite the expression.

$a = - 1 \cdot 5 - 2 (- \frac{y}{2}) + 6$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 16 Z = 53$

$a = - 1 \cdot 5 - 2 (- \frac{y}{2}) + 6$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 16 Z = 53$

Step 5.2.1.1.3

Multiply $- 1$ by $5$ .

$a = - 5 - 2 (- \frac{y}{2}) + 6$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 16 Z = 53$

Step 5.2.1.1.4

Cancel the common factor of $2$ .

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Step 5.2.1.1.4.1

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

$a = - 5 - 2 \frac{- y}{2} + 6$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 16 Z = 53$

Step 5.2.1.1.4.2

Factor $2$ out of $- 2$ .

$a = - 5 + 2 (- 1) (\frac{- y}{2}) + 6$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 16 Z = 53$

Step 5.2.1.1.4.3

Cancel the common factor.

$a = - 5 + 2 \cdot (- 1 \frac{- y}{2}) + 6$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 16 Z = 53$

Step 5.2.1.1.4.4

Rewrite the expression.

$a = - 5 - 1 (- y) + 6$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 16 Z = 53$

$a = - 5 - 1 (- y) + 6$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 16 Z = 53$

Step 5.2.1.1.5

Multiply $- 1$ by $- 1$ .

$a = - 5 + 1 y + 6$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 16 Z = 53$

Step 5.2.1.1.6

Multiply $y$ by $1$ .

$a = - 5 + y + 6$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 16 Z = 53$

$a = - 5 + y + 6$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 16 Z = 53$

Step 5.2.1.2

Add $- 5$ and $6$ .

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 16 Z = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 16 Z = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 16 Z = 53$

Step 5.3

Replace all occurrences of $Z$ in $8 y + 16 Z = 53$ with $\frac{5}{2} - \frac{y}{2}$ .

$8 y + 16 (\frac{5}{2} - \frac{y}{2}) = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

Step 5.4

Simplify the left side.

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Step 5.4.1

Simplify $8 y + 16 (\frac{5}{2} - \frac{y}{2})$ .

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Step 5.4.1.1

Simplify each term.

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Step 5.4.1.1.1

Apply the distributive property.

$8 y + 16 (\frac{5}{2}) + 16 (- \frac{y}{2}) = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

Step 5.4.1.1.2

Cancel the common factor of $2$ .

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Step 5.4.1.1.2.1

Factor $2$ out of $16$ .

$8 y + 2 (8) (\frac{5}{2}) + 16 (- \frac{y}{2}) = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

Step 5.4.1.1.2.2

Cancel the common factor.

$8 y + 2 \cdot (8 (\frac{5}{2})) + 16 (- \frac{y}{2}) = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

Step 5.4.1.1.2.3

Rewrite the expression.

$8 y + 8 \cdot 5 + 16 (- \frac{y}{2}) = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 8 \cdot 5 + 16 (- \frac{y}{2}) = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

Step 5.4.1.1.3

Multiply $8$ by $5$ .

$8 y + 40 + 16 (- \frac{y}{2}) = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

Step 5.4.1.1.4

Cancel the common factor of $2$ .

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Step 5.4.1.1.4.1

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

$8 y + 40 + 16 (\frac{- y}{2}) = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

Step 5.4.1.1.4.2

Factor $2$ out of $16$ .

$8 y + 40 + 2 (8) (\frac{- y}{2}) = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

Step 5.4.1.1.4.3

Cancel the common factor.

$8 y + 40 + 2 \cdot (8 (\frac{- y}{2})) = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

Step 5.4.1.1.4.4

Rewrite the expression.

$8 y + 40 + 8 (- y) = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 40 + 8 (- y) = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

Step 5.4.1.1.5

Multiply $- 1$ by $8$ .

$8 y + 40 - 8 y = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

$8 y + 40 - 8 y = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

Step 5.4.1.2

Combine the opposite terms in $8 y + 40 - 8 y$ .

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Step 5.4.1.2.1

Subtract $8 y$ from $8 y$ .

$0 + 40 = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

Step 5.4.1.2.2

Add $0$ and $40$ .

$40 = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

$40 = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

$40 = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

$40 = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

$40 = 53$

$a = y + 1$

$Z = \frac{5}{2} - \frac{y}{2}$

Step 6

Since $40 = 53$ is not true, there is no solution.

No solution