Finite Math Examples

$6 = 3 x + 5 y$ , $4 = 2 x - 5 y$

Step 1

Solve for $x$ in $6 = 3 x + 5 y$ .

Tap for more steps...

Step 1.1

Rewrite the equation as $3 x + 5 y = 6$ .

$3 x + 5 y = 6$

$4 = 2 x - 5 y$

Step 1.2

Subtract $5 y$ from both sides of the equation.

$3 x = 6 - 5 y$

$4 = 2 x - 5 y$

Step 1.3

Divide each term in $3 x = 6 - 5 y$ by $3$ and simplify.

Tap for more steps...

Step 1.3.1

Divide each term in $3 x = 6 - 5 y$ by $3$ .

$\frac{3 x}{3} = \frac{6}{3} + \frac{- 5 y}{3}$

$4 = 2 x - 5 y$

Step 1.3.2

Simplify the left side.

Tap for more steps...

Step 1.3.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{6}{3} + \frac{- 5 y}{3}$

$4 = 2 x - 5 y$

Step 1.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{3} + \frac{- 5 y}{3}$

$4 = 2 x - 5 y$

$x = \frac{6}{3} + \frac{- 5 y}{3}$

$4 = 2 x - 5 y$

$x = \frac{6}{3} + \frac{- 5 y}{3}$

$4 = 2 x - 5 y$

Step 1.3.3

Simplify the right side.

Tap for more steps...

Step 1.3.3.1

Simplify each term.

Tap for more steps...

Step 1.3.3.1.1

Divide $6$ by $3$ .

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

$4 = 2 x - 5 y$

Step 1.3.3.1.2

Move the negative in front of the fraction.

$x = 2 - \frac{5 y}{3}$

$4 = 2 x - 5 y$

$x = 2 - \frac{5 y}{3}$

$4 = 2 x - 5 y$

$x = 2 - \frac{5 y}{3}$

$4 = 2 x - 5 y$

$x = 2 - \frac{5 y}{3}$

$4 = 2 x - 5 y$

$x = 2 - \frac{5 y}{3}$

$4 = 2 x - 5 y$

Step 2

Replace all occurrences of $x$ with $2 - \frac{5 y}{3}$ in each equation.

Tap for more steps...

Step 2.1

Replace all occurrences of $x$ in $4 = 2 x - 5 y$ with $2 - \frac{5 y}{3}$ .

$4 = 2 (2 - \frac{5 y}{3}) - 5 y$

$x = 2 - \frac{5 y}{3}$

Step 2.2

Simplify the right side.

Tap for more steps...

Step 2.2.1

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

Tap for more steps...

Step 2.2.1.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1.1

Apply the distributive property.

$4 = 2 \cdot 2 + 2 (- \frac{5 y}{3}) - 5 y$

$x = 2 - \frac{5 y}{3}$

Step 2.2.1.1.2

Multiply $2$ by $2$ .

$4 = 4 + 2 (- \frac{5 y}{3}) - 5 y$

$x = 2 - \frac{5 y}{3}$

Step 2.2.1.1.3

Multiply $2 (- \frac{5 y}{3})$ .

Tap for more steps...

Step 2.2.1.1.3.1

Multiply $- 1$ by $2$ .

$4 = 4 - 2 \frac{5 y}{3} - 5 y$

$x = 2 - \frac{5 y}{3}$

Step 2.2.1.1.3.2

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

$4 = 4 + \frac{- 2 (5 y)}{3} - 5 y$

$x = 2 - \frac{5 y}{3}$

Step 2.2.1.1.3.3

Multiply $5$ by $- 2$ .

$4 = 4 + \frac{- 10 y}{3} - 5 y$

$x = 2 - \frac{5 y}{3}$

$4 = 4 + \frac{- 10 y}{3} - 5 y$

$x = 2 - \frac{5 y}{3}$

Step 2.2.1.1.4

Move the negative in front of the fraction.

$4 = 4 - \frac{10 y}{3} - 5 y$

$x = 2 - \frac{5 y}{3}$

$4 = 4 - \frac{10 y}{3} - 5 y$

$x = 2 - \frac{5 y}{3}$

Step 2.2.1.2

To write $- 5 y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$4 = 4 - \frac{10 y}{3} - 5 y \cdot \frac{3}{3}$

$x = 2 - \frac{5 y}{3}$

Step 2.2.1.3

Simplify terms.

Tap for more steps...

Step 2.2.1.3.1

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

$4 = 4 - \frac{10 y}{3} + \frac{- 5 y \cdot 3}{3}$

$x = 2 - \frac{5 y}{3}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$4 = 4 + \frac{- 10 y - 5 y \cdot 3}{3}$

$x = 2 - \frac{5 y}{3}$

$4 = 4 + \frac{- 10 y - 5 y \cdot 3}{3}$

$x = 2 - \frac{5 y}{3}$

Step 2.2.1.4

Simplify each term.

Tap for more steps...

Step 2.2.1.4.1

Simplify the numerator.

Tap for more steps...

Step 2.2.1.4.1.1

Factor $5 y$ out of $- 10 y - 5 y \cdot 3$ .

Tap for more steps...

Step 2.2.1.4.1.1.1

Factor $5 y$ out of $- 10 y$ .

$4 = 4 + \frac{5 y (- 2) - 5 y \cdot 3}{3}$

$x = 2 - \frac{5 y}{3}$

Step 2.2.1.4.1.1.2

Factor $5 y$ out of $- 5 y \cdot 3$ .

$4 = 4 + \frac{5 y (- 2) + 5 y (- 1 \cdot 3)}{3}$

$x = 2 - \frac{5 y}{3}$

Step 2.2.1.4.1.1.3

Factor $5 y$ out of $5 y (- 2) + 5 y (- 1 \cdot 3)$ .

$4 = 4 + \frac{5 y (- 2 - 1 \cdot 3)}{3}$

$x = 2 - \frac{5 y}{3}$

$4 = 4 + \frac{5 y (- 2 - 1 \cdot 3)}{3}$

$x = 2 - \frac{5 y}{3}$

Step 2.2.1.4.1.2

Multiply $- 1$ by $3$ .

$4 = 4 + \frac{5 y (- 2 - 3)}{3}$

$x = 2 - \frac{5 y}{3}$

Step 2.2.1.4.1.3

Subtract $3$ from $- 2$ .

$4 = 4 + \frac{5 y \cdot - 5}{3}$

$x = 2 - \frac{5 y}{3}$

Step 2.2.1.4.1.4

Multiply $- 5$ by $5$ .

$4 = 4 + \frac{- 25 y}{3}$

$x = 2 - \frac{5 y}{3}$

$4 = 4 + \frac{- 25 y}{3}$

$x = 2 - \frac{5 y}{3}$

Step 2.2.1.4.2

Move the negative in front of the fraction.

$4 = 4 - \frac{25 y}{3}$

$x = 2 - \frac{5 y}{3}$

$4 = 4 - \frac{25 y}{3}$

$x = 2 - \frac{5 y}{3}$

$4 = 4 - \frac{25 y}{3}$

$x = 2 - \frac{5 y}{3}$

$4 = 4 - \frac{25 y}{3}$

$x = 2 - \frac{5 y}{3}$

$4 = 4 - \frac{25 y}{3}$

$x = 2 - \frac{5 y}{3}$

Step 3

Solve for $y$ in $4 = 4 - \frac{25 y}{3}$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $4 - \frac{25 y}{3} = 4$ .

$4 - \frac{25 y}{3} = 4$

$x = 2 - \frac{5 y}{3}$

Step 3.2

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

Tap for more steps...

Step 3.2.1

Subtract $4$ from both sides of the equation.

$- \frac{25 y}{3} = 4 - 4$

$x = 2 - \frac{5 y}{3}$

Step 3.2.2

Subtract $4$ from $4$ .

$- \frac{25 y}{3} = 0$

$x = 2 - \frac{5 y}{3}$

$- \frac{25 y}{3} = 0$

$x = 2 - \frac{5 y}{3}$

Step 3.3

Set the numerator equal to zero.

$25 y = 0$

$x = 2 - \frac{5 y}{3}$

Step 3.4

Divide each term in $25 y = 0$ by $25$ and simplify.

Tap for more steps...

Step 3.4.1

Divide each term in $25 y = 0$ by $25$ .

$\frac{25 y}{25} = \frac{0}{25}$

$x = 2 - \frac{5 y}{3}$

Step 3.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.1

Cancel the common factor of $25$ .

Tap for more steps...

Step 3.4.2.1.1

Cancel the common factor.

$\frac{25 y}{25} = \frac{0}{25}$

$x = 2 - \frac{5 y}{3}$

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{0}{25}$

$x = 2 - \frac{5 y}{3}$

$y = \frac{0}{25}$

$x = 2 - \frac{5 y}{3}$

$y = \frac{0}{25}$

$x = 2 - \frac{5 y}{3}$

Step 3.4.3

Simplify the right side.

Tap for more steps...

Step 3.4.3.1

Divide $0$ by $25$ .

$y = 0$

$x = 2 - \frac{5 y}{3}$

$y = 0$

$x = 2 - \frac{5 y}{3}$

$y = 0$

$x = 2 - \frac{5 y}{3}$

$y = 0$

$x = 2 - \frac{5 y}{3}$

Step 4

Replace all occurrences of $y$ with $0$ in each equation.

Tap for more steps...

Step 4.1

Replace all occurrences of $y$ in $x = 2 - \frac{5 y}{3}$ with $0$ .

$x = 2 - \frac{5 (0)}{3}$

$y = 0$

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Simplify $2 - \frac{5 (0)}{3}$ .

Tap for more steps...

Step 4.2.1.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1.1

Cancel the common factor of $0$ and $3$ .

Tap for more steps...

Step 4.2.1.1.1.1

Factor $3$ out of $5 (0)$ .

$x = 2 - \frac{3 (5 \cdot (0))}{3}$

$y = 0$

Step 4.2.1.1.1.2

Cancel the common factors.

Tap for more steps...

Step 4.2.1.1.1.2.1

Factor $3$ out of $3$ .

$x = 2 - \frac{3 (5 \cdot (0))}{3 (1)}$

$y = 0$

Step 4.2.1.1.1.2.2

Cancel the common factor.

$x = 2 - \frac{3 (5 \cdot (0))}{3 \cdot 1}$

$y = 0$

Step 4.2.1.1.1.2.3

Rewrite the expression.

$x = 2 - \frac{5 \cdot (0)}{1}$

$y = 0$

Step 4.2.1.1.1.2.4

Divide $5 \cdot (0)$ by $1$ .

$x = 2 - (5 \cdot (0))$

$y = 0$

$x = 2 - (5 \cdot (0))$

$y = 0$

$x = 2 - (5 \cdot (0))$

$y = 0$

Step 4.2.1.1.2

Multiply $- (5 \cdot (0))$ .

Tap for more steps...

Step 4.2.1.1.2.1

Multiply $5$ by $0$ .

$x = 2 - 0$

$y = 0$

Step 4.2.1.1.2.2

Multiply $- 1$ by $0$ .

$x = 2 + 0$

$y = 0$

$x = 2 + 0$

$y = 0$

$x = 2 + 0$

$y = 0$

Step 4.2.1.2

Add $2$ and $0$ .

$x = 2$

$y = 0$

$x = 2$

$y = 0$

$x = 2$

$y = 0$

$x = 2$

$y = 0$

Step 5

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

$(2, 0)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(2, 0)$

Equation Form:

$x = 2, y = 0$

Step 7