Finite Math Examples

$x - 2 y = 1$ , $4 x + 4 y = 2$ , $10 x + 8 y = 4$

Step 1

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

$x = 1 + 2 y$

$4 x + 4 y = 2$

$10 x + 8 y = 4$

Step 2

Replace all occurrences of $x$ with $1 + 2 y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $4 x + 4 y = 2$ with $1 + 2 y$ .

$4 (1 + 2 y) + 4 y = 2$

$x = 1 + 2 y$

$10 x + 8 y = 4$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $4 (1 + 2 y) + 4 y$ .

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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.

$4 \cdot 1 + 4 (2 y) + 4 y = 2$

$x = 1 + 2 y$

$10 x + 8 y = 4$

Step 2.2.1.1.2

Multiply $4$ by $1$ .

$4 + 4 (2 y) + 4 y = 2$

$x = 1 + 2 y$

$10 x + 8 y = 4$

Step 2.2.1.1.3

Multiply $2$ by $4$ .

$4 + 8 y + 4 y = 2$

$x = 1 + 2 y$

$10 x + 8 y = 4$

$4 + 8 y + 4 y = 2$

$x = 1 + 2 y$

$10 x + 8 y = 4$

Step 2.2.1.2

Add $8 y$ and $4 y$ .

$4 + 12 y = 2$

$x = 1 + 2 y$

$10 x + 8 y = 4$

$4 + 12 y = 2$

$x = 1 + 2 y$

$10 x + 8 y = 4$

$4 + 12 y = 2$

$x = 1 + 2 y$

$10 x + 8 y = 4$

Step 2.3

Replace all occurrences of $x$ in $10 x + 8 y = 4$ with $1 + 2 y$ .

$10 (1 + 2 y) + 8 y = 4$

$4 + 12 y = 2$

$x = 1 + 2 y$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $10 (1 + 2 y) + 8 y$ .

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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.

$10 \cdot 1 + 10 (2 y) + 8 y = 4$

$4 + 12 y = 2$

$x = 1 + 2 y$

Step 2.4.1.1.2

Multiply $10$ by $1$ .

$10 + 10 (2 y) + 8 y = 4$

$4 + 12 y = 2$

$x = 1 + 2 y$

Step 2.4.1.1.3

Multiply $2$ by $10$ .

$10 + 20 y + 8 y = 4$

$4 + 12 y = 2$

$x = 1 + 2 y$

$10 + 20 y + 8 y = 4$

$4 + 12 y = 2$

$x = 1 + 2 y$

Step 2.4.1.2

Add $20 y$ and $8 y$ .

$10 + 28 y = 4$

$4 + 12 y = 2$

$x = 1 + 2 y$

$10 + 28 y = 4$

$4 + 12 y = 2$

$x = 1 + 2 y$

$10 + 28 y = 4$

$4 + 12 y = 2$

$x = 1 + 2 y$

$10 + 28 y = 4$

$4 + 12 y = 2$

$x = 1 + 2 y$

Step 3

Solve for $y$ in $10 + 28 y = 4$ .

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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 $10$ from both sides of the equation.

$28 y = 4 - 10$

$4 + 12 y = 2$

$x = 1 + 2 y$

Step 3.1.2

Subtract $10$ from $4$ .

$28 y = - 6$

$4 + 12 y = 2$

$x = 1 + 2 y$

$28 y = - 6$

$4 + 12 y = 2$

$x = 1 + 2 y$

Step 3.2

Divide each term in $28 y = - 6$ by $28$ and simplify.

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Step 3.2.1

Divide each term in $28 y = - 6$ by $28$ .

$\frac{28 y}{28} = \frac{- 6}{28}$

$4 + 12 y = 2$

$x = 1 + 2 y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $28$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{28 y}{28} = \frac{- 6}{28}$

$4 + 12 y = 2$

$x = 1 + 2 y$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 6}{28}$

$4 + 12 y = 2$

$x = 1 + 2 y$

$y = \frac{- 6}{28}$

$4 + 12 y = 2$

$x = 1 + 2 y$

$y = \frac{- 6}{28}$

$4 + 12 y = 2$

$x = 1 + 2 y$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $- 6$ and $28$ .

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Step 3.2.3.1.1

Factor $2$ out of $- 6$ .

$y = \frac{2 (- 3)}{28}$

$4 + 12 y = 2$

$x = 1 + 2 y$

Step 3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.1.2.1

Factor $2$ out of $28$ .

$y = \frac{2 \cdot - 3}{2 \cdot 14}$

$4 + 12 y = 2$

$x = 1 + 2 y$

Step 3.2.3.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot - 3}{2 \cdot 14}$

$4 + 12 y = 2$

$x = 1 + 2 y$

Step 3.2.3.1.2.3

Rewrite the expression.

$y = \frac{- 3}{14}$

$4 + 12 y = 2$

$x = 1 + 2 y$

$y = \frac{- 3}{14}$

$4 + 12 y = 2$

$x = 1 + 2 y$

$y = \frac{- 3}{14}$

$4 + 12 y = 2$

$x = 1 + 2 y$

Step 3.2.3.2

Move the negative in front of the fraction.

$y = - \frac{3}{14}$

$4 + 12 y = 2$

$x = 1 + 2 y$

$y = - \frac{3}{14}$

$4 + 12 y = 2$

$x = 1 + 2 y$

$y = - \frac{3}{14}$

$4 + 12 y = 2$

$x = 1 + 2 y$

$y = - \frac{3}{14}$

$4 + 12 y = 2$

$x = 1 + 2 y$

Step 4

Replace all occurrences of $y$ with $- \frac{3}{14}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $4 + 12 y = 2$ with $- \frac{3}{14}$ .

$4 + 12 (- \frac{3}{14}) = 2$

$y = - \frac{3}{14}$

$x = 1 + 2 y$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $4 + 12 (- \frac{3}{14})$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Cancel the common factor of $2$ .

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Step 4.2.1.1.1.1

Move the leading negative in $- \frac{3}{14}$ into the numerator.

$4 + 12 (\frac{- 3}{14}) = 2$

$y = - \frac{3}{14}$

$x = 1 + 2 y$

Step 4.2.1.1.1.2

Factor $2$ out of $12$ .

$4 + 2 (6) (\frac{- 3}{14}) = 2$

$y = - \frac{3}{14}$

$x = 1 + 2 y$

Step 4.2.1.1.1.3

Factor $2$ out of $14$ .

$4 + 2 \cdot (6 (\frac{- 3}{2 \cdot 7})) = 2$

$y = - \frac{3}{14}$

$x = 1 + 2 y$

Step 4.2.1.1.1.4

Cancel the common factor.

$4 + 2 \cdot (6 (\frac{- 3}{2 \cdot 7})) = 2$

$y = - \frac{3}{14}$

$x = 1 + 2 y$

Step 4.2.1.1.1.5

Rewrite the expression.

$4 + 6 (\frac{- 3}{7}) = 2$

$y = - \frac{3}{14}$

$x = 1 + 2 y$

$4 + 6 (\frac{- 3}{7}) = 2$

$y = - \frac{3}{14}$

$x = 1 + 2 y$

Step 4.2.1.1.2

Combine $6$ and $\frac{- 3}{7}$ .

$4 + \frac{6 \cdot - 3}{7} = 2$

$y = - \frac{3}{14}$

$x = 1 + 2 y$

Step 4.2.1.1.3

Multiply $6$ by $- 3$ .

$4 + \frac{- 18}{7} = 2$

$y = - \frac{3}{14}$

$x = 1 + 2 y$

Step 4.2.1.1.4

Move the negative in front of the fraction.

$4 - \frac{18}{7} = 2$

$y = - \frac{3}{14}$

$x = 1 + 2 y$

$4 - \frac{18}{7} = 2$

$y = - \frac{3}{14}$

$x = 1 + 2 y$

Step 4.2.1.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$4 \cdot \frac{7}{7} - \frac{18}{7} = 2$

$y = - \frac{3}{14}$

$x = 1 + 2 y$

Step 4.2.1.3

Combine $4$ and $\frac{7}{7}$ .

$\frac{4 \cdot 7}{7} - \frac{18}{7} = 2$

$y = - \frac{3}{14}$

$x = 1 + 2 y$

Step 4.2.1.4

Combine the numerators over the common denominator.

$\frac{4 \cdot 7 - 18}{7} = 2$

$y = - \frac{3}{14}$

$x = 1 + 2 y$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $4$ by $7$ .

$\frac{28 - 18}{7} = 2$

$y = - \frac{3}{14}$

$x = 1 + 2 y$

Step 4.2.1.5.2

Subtract $18$ from $28$ .

$\frac{10}{7} = 2$

$y = - \frac{3}{14}$

$x = 1 + 2 y$

$\frac{10}{7} = 2$

$y = - \frac{3}{14}$

$x = 1 + 2 y$

$\frac{10}{7} = 2$

$y = - \frac{3}{14}$

$x = 1 + 2 y$

$\frac{10}{7} = 2$

$y = - \frac{3}{14}$

$x = 1 + 2 y$

Step 4.3

Replace all occurrences of $y$ in $x = 1 + 2 y$ with $- \frac{3}{14}$ .

$x = 1 + 2 (- \frac{3}{14})$

$\frac{10}{7} = 2$

$y = - \frac{3}{14}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $1 + 2 (- \frac{3}{14})$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Cancel the common factor of $2$ .

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Step 4.4.1.1.1.1

Move the leading negative in $- \frac{3}{14}$ into the numerator.

$x = 1 + 2 (\frac{- 3}{14})$

$\frac{10}{7} = 2$

$y = - \frac{3}{14}$

Step 4.4.1.1.1.2

Factor $2$ out of $14$ .

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

$\frac{10}{7} = 2$

$y = - \frac{3}{14}$

Step 4.4.1.1.1.3

Cancel the common factor.

$x = 1 + 2 (\frac{- 3}{2 \cdot 7})$

$\frac{10}{7} = 2$

$y = - \frac{3}{14}$

Step 4.4.1.1.1.4

Rewrite the expression.

$x = 1 + \frac{- 3}{7}$

$\frac{10}{7} = 2$

$y = - \frac{3}{14}$

$x = 1 + \frac{- 3}{7}$

$\frac{10}{7} = 2$

$y = - \frac{3}{14}$

Step 4.4.1.1.2

Move the negative in front of the fraction.

$x = 1 - \frac{3}{7}$

$\frac{10}{7} = 2$

$y = - \frac{3}{14}$

$x = 1 - \frac{3}{7}$

$\frac{10}{7} = 2$

$y = - \frac{3}{14}$

Step 4.4.1.2

Simplify the expression.

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Step 4.4.1.2.1

Write $1$ as a fraction with a common denominator.

$x = \frac{7}{7} - \frac{3}{7}$

$\frac{10}{7} = 2$

$y = - \frac{3}{14}$

Step 4.4.1.2.2

Combine the numerators over the common denominator.

$x = \frac{7 - 3}{7}$

$\frac{10}{7} = 2$

$y = - \frac{3}{14}$

Step 4.4.1.2.3

Subtract $3$ from $7$ .

$x = \frac{4}{7}$

$\frac{10}{7} = 2$

$y = - \frac{3}{14}$

$x = \frac{4}{7}$

$\frac{10}{7} = 2$

$y = - \frac{3}{14}$

$x = \frac{4}{7}$

$\frac{10}{7} = 2$

$y = - \frac{3}{14}$

$x = \frac{4}{7}$

$\frac{10}{7} = 2$

$y = - \frac{3}{14}$

$x = \frac{4}{7}$

$\frac{10}{7} = 2$

$y = - \frac{3}{14}$

Step 5

Since $\frac{10}{7} = 2$ is not true, there is no solution.

No solution

Step 6