Finite Math Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Add $3 y$ to both sides of the equation.

$5 x = - 6 + 3 y$

$6 x + 2 y = - 3$

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

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

$6 x + 2 y = - 3$

Step 1.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 1.2.2.1.1

Cancel the common factor.

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

$6 x + 2 y = - 3$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$6 x + 2 y = - 3$

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

$6 x + 2 y = - 3$

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

$6 x + 2 y = - 3$

Step 1.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.1

Move the negative in front of the fraction.

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

$6 x + 2 y = - 3$

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

$6 x + 2 y = - 3$

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

$6 x + 2 y = - 3$

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

$6 x + 2 y = - 3$

Step 2

Replace all occurrences of $x$ with $- \frac{6}{5} + \frac{3 y}{5}$ in each equation.

Tap for more steps...

Step 2.1

Replace all occurrences of $x$ in $6 x + 2 y = - 3$ with $- \frac{6}{5} + \frac{3 y}{5}$ .

$6 (- \frac{6}{5} + \frac{3 y}{5}) + 2 y = - 3$

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

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify $6 (- \frac{6}{5} + \frac{3 y}{5}) + 2 y$ .

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.

$6 (- \frac{6}{5}) + 6 (\frac{3 y}{5}) + 2 y = - 3$

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

Step 2.2.1.1.2

Multiply $6 (- \frac{6}{5})$ .

Tap for more steps...

Step 2.2.1.1.2.1

Multiply $- 1$ by $6$ .

$- 6 (\frac{6}{5}) + 6 (\frac{3 y}{5}) + 2 y = - 3$

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

Step 2.2.1.1.2.2

Combine $- 6$ and $\frac{6}{5}$ .

$\frac{- 6 \cdot 6}{5} + 6 (\frac{3 y}{5}) + 2 y = - 3$

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

Step 2.2.1.1.2.3

Multiply $- 6$ by $6$ .

$\frac{- 36}{5} + 6 (\frac{3 y}{5}) + 2 y = - 3$

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

$\frac{- 36}{5} + 6 (\frac{3 y}{5}) + 2 y = - 3$

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

Step 2.2.1.1.3

Multiply $6 \frac{3 y}{5}$ .

Tap for more steps...

Step 2.2.1.1.3.1

Combine $6$ and $\frac{3 y}{5}$ .

$\frac{- 36}{5} + \frac{6 (3 y)}{5} + 2 y = - 3$

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

Step 2.2.1.1.3.2

Multiply $3$ by $6$ .

$\frac{- 36}{5} + \frac{18 y}{5} + 2 y = - 3$

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

$\frac{- 36}{5} + \frac{18 y}{5} + 2 y = - 3$

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

Step 2.2.1.1.4

Move the negative in front of the fraction.

$- \frac{36}{5} + \frac{18 y}{5} + 2 y = - 3$

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

$- \frac{36}{5} + \frac{18 y}{5} + 2 y = - 3$

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

Step 2.2.1.2

To write $2 y$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{36}{5} + \frac{18 y}{5} + 2 y \cdot \frac{5}{5} = - 3$

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

Step 2.2.1.3

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

$- \frac{36}{5} + \frac{18 y}{5} + \frac{2 y \cdot 5}{5} = - 3$

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

Step 2.2.1.4

Combine the numerators over the common denominator.

$- \frac{36}{5} + \frac{18 y + 2 y \cdot 5}{5} = - 3$

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

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{- 36 + 18 y + 2 y \cdot 5}{5} = - 3$

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

Step 2.2.1.6

Multiply $5$ by $2$ .

$\frac{- 36 + 18 y + 10 y}{5} = - 3$

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

Step 2.2.1.7

Add $18 y$ and $10 y$ .

$\frac{- 36 + 28 y}{5} = - 3$

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

Step 2.2.1.8

Factor $4$ out of $- 36 + 28 y$ .

Tap for more steps...

Step 2.2.1.8.1

Factor $4$ out of $- 36$ .

$\frac{4 (- 9) + 28 y}{5} = - 3$

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

Step 2.2.1.8.2

Factor $4$ out of $28 y$ .

$\frac{4 (- 9) + 4 (7 y)}{5} = - 3$

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

Step 2.2.1.8.3

Factor $4$ out of $4 (- 9) + 4 (7 y)$ .

$\frac{4 (- 9 + 7 y)}{5} = - 3$

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

$\frac{4 (- 9 + 7 y)}{5} = - 3$

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

Step 2.2.1.9

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{4 (- 1 \cdot 9 + 7 y)}{5} = - 3$

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

Step 2.2.1.10

Factor $- 1$ out of $7 y$ .

$\frac{4 (- 1 \cdot 9 - (- 7 y))}{5} = - 3$

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

Step 2.2.1.11

Factor $- 1$ out of $- 1 (9) - (- 7 y)$ .

$\frac{4 (- 1 (9 - 7 y))}{5} = - 3$

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

Step 2.2.1.12

Move the negative in front of the fraction.

$- \frac{4 (9 - 7 y)}{5} = - 3$

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

$- \frac{4 (9 - 7 y)}{5} = - 3$

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

$- \frac{4 (9 - 7 y)}{5} = - 3$

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

$- \frac{4 (9 - 7 y)}{5} = - 3$

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

Step 3

Solve for $y$ in $- \frac{4 (9 - 7 y)}{5} = - 3$ .

Tap for more steps...

Step 3.1

Multiply both sides of the equation by $- \frac{5}{4}$ .

$- \frac{5}{4} \cdot (- \frac{4 (9 - 7 y)}{5}) = - \frac{5}{4} \cdot - 3$

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

Step 3.2

Simplify both sides of the equation.

Tap for more steps...

Step 3.2.1

Simplify the left side.

Tap for more steps...

Step 3.2.1.1

Simplify $- \frac{5}{4} (- \frac{4 (9 - 7 y)}{5})$ .

Tap for more steps...

Step 3.2.1.1.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.2.1.1.1.1

Move the leading negative in $- \frac{5}{4}$ into the numerator.

$\frac{- 5}{4} \cdot (- \frac{4 (9 - 7 y)}{5}) = - \frac{5}{4} \cdot - 3$

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

Step 3.2.1.1.1.2

Move the leading negative in $- \frac{4 (9 - 7 y)}{5}$ into the numerator.

$\frac{- 5}{4} \cdot \frac{- 4 (9 - 7 y)}{5} = - \frac{5}{4} \cdot - 3$

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

Step 3.2.1.1.1.3

Factor $5$ out of $- 5$ .

$\frac{5 (- 1)}{4} \cdot \frac{- 4 (9 - 7 y)}{5} = - \frac{5}{4} \cdot - 3$

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

Step 3.2.1.1.1.4

Cancel the common factor.

$\frac{5 \cdot - 1}{4} \cdot \frac{- 4 (9 - 7 y)}{5} = - \frac{5}{4} \cdot - 3$

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

Step 3.2.1.1.1.5

Rewrite the expression.

$\frac{- 1}{4} \cdot (- 4 (9 - 7 y)) = - \frac{5}{4} \cdot - 3$

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

$\frac{- 1}{4} \cdot (- 4 (9 - 7 y)) = - \frac{5}{4} \cdot - 3$

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

Step 3.2.1.1.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.2.1.1.2.1

Factor $4$ out of $- 4 (9 - 7 y)$ .

$\frac{- 1}{4} \cdot (4 (- (9 - 7 y))) = - \frac{5}{4} \cdot - 3$

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

Step 3.2.1.1.2.2

Cancel the common factor.

$\frac{- 1}{4} \cdot (4 (- (9 - 7 y))) = - \frac{5}{4} \cdot - 3$

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

Step 3.2.1.1.2.3

Rewrite the expression.

$9 - 7 y = - \frac{5}{4} \cdot - 3$

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

$9 - 7 y = - \frac{5}{4} \cdot - 3$

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

Step 3.2.1.1.3

Multiply.

Tap for more steps...

Step 3.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 (9 - 7 y) = - \frac{5}{4} \cdot - 3$

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

Step 3.2.1.1.3.2

Multiply $9 - 7 y$ by $1$ .

$9 - 7 y = - \frac{5}{4} \cdot - 3$

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

$9 - 7 y = - \frac{5}{4} \cdot - 3$

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

$9 - 7 y = - \frac{5}{4} \cdot - 3$

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

$9 - 7 y = - \frac{5}{4} \cdot - 3$

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

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Multiply $- \frac{5}{4} \cdot - 3$ .

Tap for more steps...

Step 3.2.2.1.1

Multiply $- 3$ by $- 1$ .

$9 - 7 y = 3 (\frac{5}{4})$

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

Step 3.2.2.1.2

Combine $3$ and $\frac{5}{4}$ .

$9 - 7 y = \frac{3 \cdot 5}{4}$

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

Step 3.2.2.1.3

Multiply $3$ by $5$ .

$9 - 7 y = \frac{15}{4}$

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

$9 - 7 y = \frac{15}{4}$

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

$9 - 7 y = \frac{15}{4}$

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

$9 - 7 y = \frac{15}{4}$

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

Step 3.3

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

Tap for more steps...

Step 3.3.1

Subtract $9$ from both sides of the equation.

$- 7 y = \frac{15}{4} - 9$

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

Step 3.3.2

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- 7 y = \frac{15}{4} - 9 \cdot \frac{4}{4}$

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

Step 3.3.3

Combine $- 9$ and $\frac{4}{4}$ .

$- 7 y = \frac{15}{4} + \frac{- 9 \cdot 4}{4}$

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

Step 3.3.4

Combine the numerators over the common denominator.

$- 7 y = \frac{15 - 9 \cdot 4}{4}$

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

Step 3.3.5

Simplify the numerator.

Tap for more steps...

Step 3.3.5.1

Multiply $- 9$ by $4$ .

$- 7 y = \frac{15 - 36}{4}$

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

Step 3.3.5.2

Subtract $36$ from $15$ .

$- 7 y = \frac{- 21}{4}$

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

$- 7 y = \frac{- 21}{4}$

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

Step 3.3.6

Move the negative in front of the fraction.

$- 7 y = - \frac{21}{4}$

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

$- 7 y = - \frac{21}{4}$

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

Step 3.4

Divide each term in $- 7 y = - \frac{21}{4}$ by $- 7$ and simplify.

Tap for more steps...

Step 3.4.1

Divide each term in $- 7 y = - \frac{21}{4}$ by $- 7$ .

$\frac{- 7 y}{- 7} = \frac{- \frac{21}{4}}{- 7}$

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

Step 3.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.1

Cancel the common factor of $- 7$ .

Tap for more steps...

Step 3.4.2.1.1

Cancel the common factor.

$\frac{- 7 y}{- 7} = \frac{- \frac{21}{4}}{- 7}$

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

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{21}{4}}{- 7}$

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

$y = \frac{- \frac{21}{4}}{- 7}$

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

$y = \frac{- \frac{21}{4}}{- 7}$

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

Step 3.4.3

Simplify the right side.

Tap for more steps...

Step 3.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{21}{4} \cdot \frac{1}{- 7}$

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

Step 3.4.3.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 3.4.3.2.1

Move the leading negative in $- \frac{21}{4}$ into the numerator.

$y = \frac{- 21}{4} \cdot \frac{1}{- 7}$

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

Step 3.4.3.2.2

Factor $7$ out of $- 21$ .

$y = \frac{7 (- 3)}{4} \cdot \frac{1}{- 7}$

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

Step 3.4.3.2.3

Factor $7$ out of $- 7$ .

$y = \frac{7 \cdot - 3}{4} \cdot \frac{1}{7 \cdot - 1}$

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

Step 3.4.3.2.4

Cancel the common factor.

$y = \frac{7 \cdot - 3}{4} \cdot \frac{1}{7 \cdot - 1}$

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

Step 3.4.3.2.5

Rewrite the expression.

$y = \frac{- 3}{4} \cdot \frac{1}{- 1}$

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

$y = \frac{- 3}{4} \cdot \frac{1}{- 1}$

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

Step 3.4.3.3

Multiply $\frac{- 3}{4}$ by $\frac{1}{- 1}$ .

$y = \frac{- 3}{4 \cdot - 1}$

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

Step 3.4.3.4

Multiply $4$ by $- 1$ .

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

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

Step 3.4.3.5

Dividing two negative values results in a positive value.

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

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

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

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

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

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

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

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

Step 4

Replace all occurrences of $y$ with $\frac{3}{4}$ in each equation.

Tap for more steps...

Step 4.1

Replace all occurrences of $y$ in $x = - \frac{6}{5} + \frac{3 y}{5}$ with $\frac{3}{4}$ .

$x = - \frac{6}{5} + \frac{3 (\frac{3}{4})}{5}$

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

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Simplify $- \frac{6}{5} + \frac{3 (\frac{3}{4})}{5}$ .

Tap for more steps...

Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{- 6 + 3 (\frac{3}{4})}{5}$

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

Step 4.2.1.2

Multiply $3 (\frac{3}{4})$ .

Tap for more steps...

Step 4.2.1.2.1

Combine $3$ and $\frac{3}{4}$ .

$x = \frac{- 6 + \frac{3 \cdot 3}{4}}{5}$

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

Step 4.2.1.2.2

Multiply $3$ by $3$ .

$x = \frac{- 6 + \frac{9}{4}}{5}$

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

$x = \frac{- 6 + \frac{9}{4}}{5}$

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

Step 4.2.1.3

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{- 6 \cdot \frac{4}{4} + \frac{9}{4}}{5}$

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

Step 4.2.1.4

Combine $- 6$ and $\frac{4}{4}$ .

$x = \frac{\frac{- 6 \cdot 4}{4} + \frac{9}{4}}{5}$

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

Step 4.2.1.5

Combine the numerators over the common denominator.

$x = \frac{\frac{- 6 \cdot 4 + 9}{4}}{5}$

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

Step 4.2.1.6

Simplify the numerator.

Tap for more steps...

Step 4.2.1.6.1

Multiply $- 6$ by $4$ .

$x = \frac{\frac{- 24 + 9}{4}}{5}$

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

Step 4.2.1.6.2

Add $- 24$ and $9$ .

$x = \frac{\frac{- 15}{4}}{5}$

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

$x = \frac{\frac{- 15}{4}}{5}$

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

Step 4.2.1.7

Move the negative in front of the fraction.

$x = \frac{- \frac{15}{4}}{5}$

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

Step 4.2.1.8

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{15}{4} \cdot \frac{1}{5}$

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

Step 4.2.1.9

Cancel the common factor of $5$ .

Tap for more steps...

Step 4.2.1.9.1

Move the leading negative in $- \frac{15}{4}$ into the numerator.

$x = \frac{- 15}{4} \cdot \frac{1}{5}$

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

Step 4.2.1.9.2

Factor $5$ out of $- 15$ .

$x = \frac{5 (- 3)}{4} \cdot \frac{1}{5}$

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

Step 4.2.1.9.3

Cancel the common factor.

$x = \frac{5 \cdot - 3}{4} \cdot \frac{1}{5}$

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

Step 4.2.1.9.4

Rewrite the expression.

$x = \frac{- 3}{4}$

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

$x = \frac{- 3}{4}$

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

Step 4.2.1.10

Move the negative in front of the fraction.

$x = - \frac{3}{4}$

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

$x = - \frac{3}{4}$

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

$x = - \frac{3}{4}$

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

$x = - \frac{3}{4}$

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

Step 5

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

$(- \frac{3}{4}, \frac{3}{4})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- \frac{3}{4}, \frac{3}{4})$

Equation Form:

$x = - \frac{3}{4}, y = \frac{3}{4}$

Step 7