Finite Math Examples

$3 x - 2 y = - 3$ , $5 x - y = 2$

Step 1

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

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Step 1.1

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

$3 x = - 3 + 2 y$

$5 x - y = 2$

Step 1.2

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

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Step 1.2.1

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

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

$5 x - y = 2$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

$5 x - y = 2$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$5 x - y = 2$

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

$5 x - y = 2$

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

$5 x - y = 2$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $- 3$ by $3$ .

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

$5 x - y = 2$

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

$5 x - y = 2$

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

$5 x - y = 2$

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

$5 x - y = 2$

Step 2

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

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Step 2.1

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

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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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.

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

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

Step 2.2.1.1.2

Multiply $5$ by $- 1$ .

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

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

Step 2.2.1.1.3

Multiply $5 \frac{2 y}{3}$ .

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Step 2.2.1.1.3.1

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

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

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

Step 2.2.1.1.3.2

Multiply $2$ by $5$ .

$- 5 + \frac{10 y}{3} - y = 2$

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

$- 5 + \frac{10 y}{3} - y = 2$

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

$- 5 + \frac{10 y}{3} - y = 2$

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

Step 2.2.1.2

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

$- 5 + \frac{10 y}{3} - y \cdot \frac{3}{3} = 2$

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

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

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

$- 5 + \frac{10 y}{3} + \frac{- y \cdot 3}{3} = 2$

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

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$- 5 + \frac{10 y - y \cdot 3}{3} = 2$

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

$- 5 + \frac{10 y - y \cdot 3}{3} = 2$

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

Step 2.2.1.4

Simplify each term.

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Step 2.2.1.4.1

Simplify the numerator.

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Step 2.2.1.4.1.1

Factor $y$ out of $10 y - y \cdot 3$ .

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Step 2.2.1.4.1.1.1

Factor $y$ out of $10 y$ .

$- 5 + \frac{y \cdot 10 - y \cdot 3}{3} = 2$

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

Step 2.2.1.4.1.1.2

Factor $y$ out of $- y \cdot 3$ .

$- 5 + \frac{y \cdot 10 + y (- 1 \cdot 3)}{3} = 2$

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

Step 2.2.1.4.1.1.3

Factor $y$ out of $y \cdot 10 + y (- 1 \cdot 3)$ .

$- 5 + \frac{y (10 - 1 \cdot 3)}{3} = 2$

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

$- 5 + \frac{y (10 - 1 \cdot 3)}{3} = 2$

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

Step 2.2.1.4.1.2

Multiply $- 1$ by $3$ .

$- 5 + \frac{y (10 - 3)}{3} = 2$

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

Step 2.2.1.4.1.3

Subtract $3$ from $10$ .

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

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

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

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

Step 2.2.1.4.2

Move $7$ to the left of $y$ .

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

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

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

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

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

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

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

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

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

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

Step 3

Solve for $y$ in $- 5 + \frac{7 y}{3} = 2$ .

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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 $5$ to both sides of the equation.

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

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

Step 3.1.2

Add $2$ and $5$ .

$\frac{7 y}{3} = 7$

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

$\frac{7 y}{3} = 7$

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

Step 3.2

Multiply both sides of the equation by $\frac{3}{7}$ .

$\frac{3}{7} \cdot \frac{7 y}{3} = \frac{3}{7} \cdot 7$

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

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $\frac{3}{7} \cdot \frac{7 y}{3}$ .

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Step 3.3.1.1.1

Cancel the common factor of $3$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{3}{7} \cdot \frac{7 y}{3} = \frac{3}{7} \cdot 7$

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

Step 3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{7} \cdot (7 y) = \frac{3}{7} \cdot 7$

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

$\frac{1}{7} \cdot (7 y) = \frac{3}{7} \cdot 7$

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

Step 3.3.1.1.2

Cancel the common factor of $7$ .

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Step 3.3.1.1.2.1

Factor $7$ out of $7 y$ .

$\frac{1}{7} \cdot (7 (y)) = \frac{3}{7} \cdot 7$

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

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{1}{7} \cdot (7 y) = \frac{3}{7} \cdot 7$

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

Step 3.3.1.1.2.3

Rewrite the expression.

$y = \frac{3}{7} \cdot 7$

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

$y = \frac{3}{7} \cdot 7$

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

$y = \frac{3}{7} \cdot 7$

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

$y = \frac{3}{7} \cdot 7$

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

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Cancel the common factor of $7$ .

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Step 3.3.2.1.1

Cancel the common factor.

$y = \frac{3}{7} \cdot 7$

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

Step 3.3.2.1.2

Rewrite the expression.

$y = 3$

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

$y = 3$

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

$y = 3$

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

$y = 3$

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

$y = 3$

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

Step 4

Replace all occurrences of $y$ with $3$ in each equation.

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Step 4.1

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

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

$y = 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

Multiply $2$ by $3$ .

$x = - 1 + \frac{6}{3}$

$y = 3$

Step 4.2.1.2

Divide $6$ by $3$ .

$x = - 1 + 2$

$y = 3$

Step 4.2.1.3

Add $- 1$ and $2$ .

$x = 1$

$y = 3$

$x = 1$

$y = 3$

$x = 1$

$y = 3$

$x = 1$

$y = 3$

Step 5

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

$(1, 3)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(1, 3)$

Equation Form:

$x = 1, y = 3$

Step 7