Finite Math Examples

$2 x - 5 y = 10$ , $x + y = 2$

Step 1

Subtract $y$ from both sides of the equation.

$x = 2 - y$

$2 x - 5 y = 10$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $2 x - 5 y = 10$ with $2 - y$ .

$2 (2 - y) - 5 y = 10$

$x = 2 - y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (2 - y) - 5 y$ .

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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.

$2 \cdot 2 + 2 (- y) - 5 y = 10$

$x = 2 - y$

Step 2.2.1.1.2

Multiply $2$ by $2$ .

$4 + 2 (- y) - 5 y = 10$

$x = 2 - y$

Step 2.2.1.1.3

Multiply $- 1$ by $2$ .

$4 - 2 y - 5 y = 10$

$x = 2 - y$

$4 - 2 y - 5 y = 10$

$x = 2 - y$

Step 2.2.1.2

Subtract $5 y$ from $- 2 y$ .

$4 - 7 y = 10$

$x = 2 - y$

$4 - 7 y = 10$

$x = 2 - y$

$4 - 7 y = 10$

$x = 2 - y$

$4 - 7 y = 10$

$x = 2 - y$

Step 3

Solve for $y$ in $4 - 7 y = 10$ .

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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 $4$ from both sides of the equation.

$- 7 y = 10 - 4$

$x = 2 - y$

Step 3.1.2

Subtract $4$ from $10$ .

$- 7 y = 6$

$x = 2 - y$

$- 7 y = 6$

$x = 2 - y$

Step 3.2

Divide each term in $- 7 y = 6$ by $- 7$ and simplify.

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Step 3.2.1

Divide each term in $- 7 y = 6$ by $- 7$ .

$\frac{- 7 y}{- 7} = \frac{6}{- 7}$

$x = 2 - y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 7$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 7 y}{- 7} = \frac{6}{- 7}$

$x = 2 - y$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{6}{- 7}$

$x = 2 - y$

$y = \frac{6}{- 7}$

$x = 2 - y$

$y = \frac{6}{- 7}$

$x = 2 - y$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative in front of the fraction.

$y = - \frac{6}{7}$

$x = 2 - y$

$y = - \frac{6}{7}$

$x = 2 - y$

$y = - \frac{6}{7}$

$x = 2 - y$

$y = - \frac{6}{7}$

$x = 2 - y$

Step 4

Replace all occurrences of $y$ with $- \frac{6}{7}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 2 - y$ with $- \frac{6}{7}$ .

$x = 2 - (- \frac{6}{7})$

$y = - \frac{6}{7}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $2 - (- \frac{6}{7})$ .

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Step 4.2.1.1

Multiply $- (- \frac{6}{7})$ .

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Step 4.2.1.1.1

Multiply $- 1$ by $- 1$ .

$x = 2 + 1 (\frac{6}{7})$

$y = - \frac{6}{7}$

Step 4.2.1.1.2

Multiply $\frac{6}{7}$ by $1$ .

$x = 2 + \frac{6}{7}$

$y = - \frac{6}{7}$

$x = 2 + \frac{6}{7}$

$y = - \frac{6}{7}$

Step 4.2.1.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x = 2 \cdot \frac{7}{7} + \frac{6}{7}$

$y = - \frac{6}{7}$

Step 4.2.1.3

Combine $2$ and $\frac{7}{7}$ .

$x = \frac{2 \cdot 7}{7} + \frac{6}{7}$

$y = - \frac{6}{7}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x = \frac{2 \cdot 7 + 6}{7}$

$y = - \frac{6}{7}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $2$ by $7$ .

$x = \frac{14 + 6}{7}$

$y = - \frac{6}{7}$

Step 4.2.1.5.2

Add $14$ and $6$ .

$x = \frac{20}{7}$

$y = - \frac{6}{7}$

$x = \frac{20}{7}$

$y = - \frac{6}{7}$

$x = \frac{20}{7}$

$y = - \frac{6}{7}$

$x = \frac{20}{7}$

$y = - \frac{6}{7}$

$x = \frac{20}{7}$

$y = - \frac{6}{7}$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(\frac{20}{7}, - \frac{6}{7})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{20}{7}, - \frac{6}{7})$

Equation Form:

$x = \frac{20}{7}, y = - \frac{6}{7}$

Step 7