Algebra Examples

$\frac{1}{4} x + \frac{3}{5} y = 1$ $\frac{1}{6} x - \frac{1}{3} y = - 3$

Step 1

Solve for $x$ in $\frac{1}{4} x + \frac{3}{5} y = 1$ .

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

Combine $\frac{1}{4}$ and $x$ .

$\frac{x}{4} + \frac{3}{5} y = 1$

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

Step 1.1.2

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

$\frac{x}{4} + \frac{3 y}{5} = 1$

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

$\frac{x}{4} + \frac{3 y}{5} = 1$

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

Step 1.2

Subtract $\frac{3 y}{5}$ from both sides of the equation.

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

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

Step 1.3

Multiply both sides of the equation by $4$ .

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

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

Step 1.4

Simplify both sides of the equation.

Tap for more steps...

Step 1.4.1

Simplify the left side.

Tap for more steps...

Step 1.4.1.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.4.1.1.1

Cancel the common factor.

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

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

Step 1.4.1.1.2

Rewrite the expression.

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

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

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

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

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

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

Step 1.4.2

Simplify the right side.

Tap for more steps...

Step 1.4.2.1

Simplify $4 (1 - \frac{3 y}{5})$ .

Tap for more steps...

Step 1.4.2.1.1

Apply the distributive property.

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

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

Step 1.4.2.1.2

Multiply $4$ by $1$ .

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

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

Step 1.4.2.1.3

Multiply $4 (- \frac{3 y}{5})$ .

Tap for more steps...

Step 1.4.2.1.3.1

Multiply $- 1$ by $4$ .

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

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

Step 1.4.2.1.3.2

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

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

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

Step 1.4.2.1.3.3

Multiply $3$ by $- 4$ .

$x = 4 + \frac{- 12 y}{5}$

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

$x = 4 + \frac{- 12 y}{5}$

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

Step 1.4.2.1.4

Move the negative in front of the fraction.

$x = 4 - \frac{12 y}{5}$

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

$x = 4 - \frac{12 y}{5}$

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

$x = 4 - \frac{12 y}{5}$

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

$x = 4 - \frac{12 y}{5}$

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

Step 1.5

Reorder $4$ and $- \frac{12 y}{5}$ .

$x = - \frac{12 y}{5} + 4$

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

$x = - \frac{12 y}{5} + 4$

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

Step 2

Replace all occurrences of $x$ with $- \frac{12 y}{5} + 4$ in each equation.

Tap for more steps...

Step 2.1

Replace all occurrences of $x$ in $\frac{1}{6} x - \frac{1}{3} y = - 3$ with $- \frac{12 y}{5} + 4$ .

$\frac{1}{6} \cdot (- \frac{12 y}{5} + 4) - \frac{1}{3} y = - 3$

$x = - \frac{12 y}{5} + 4$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify $\frac{1}{6} \cdot (- \frac{12 y}{5} + 4) - \frac{1}{3} y$ .

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.

$\frac{1}{6} \cdot (- \frac{12 y}{5}) + \frac{1}{6} \cdot 4 - \frac{1}{3} y = - 3$

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.1.2

Cancel the common factor of $6$ .

Tap for more steps...

Step 2.2.1.1.2.1

Move the leading negative in $- \frac{12 y}{5}$ into the numerator.

$\frac{1}{6} \cdot \frac{- 12 y}{5} + \frac{1}{6} \cdot 4 - \frac{1}{3} y = - 3$

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.1.2.2

Factor $6$ out of $- 12 y$ .

$\frac{1}{6} \cdot \frac{6 (- 2 y)}{5} + \frac{1}{6} \cdot 4 - \frac{1}{3} y = - 3$

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.1.2.3

Cancel the common factor.

$\frac{1}{6} \cdot \frac{6 (- 2 y)}{5} + \frac{1}{6} \cdot 4 - \frac{1}{3} y = - 3$

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.1.2.4

Rewrite the expression.

$\frac{- 2 y}{5} + \frac{1}{6} \cdot 4 - \frac{1}{3} y = - 3$

$x = - \frac{12 y}{5} + 4$

$\frac{- 2 y}{5} + \frac{1}{6} \cdot 4 - \frac{1}{3} y = - 3$

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.1.3.1

Factor $2$ out of $6$ .

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

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.1.3.2

Factor $2$ out of $4$ .

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

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.1.3.3

Cancel the common factor.

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

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.1.3.4

Rewrite the expression.

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

$x = - \frac{12 y}{5} + 4$

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

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.1.4

Combine $\frac{1}{3}$ and $2$ .

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

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.1.5

Move the negative in front of the fraction.

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

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.1.6

Combine $y$ and $\frac{1}{3}$ .

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

$x = - \frac{12 y}{5} + 4$

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

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.2

To write $- \frac{2 y}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.3

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

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

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.4

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 2.2.1.4.1

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

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

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.4.2

Multiply $5$ by $3$ .

$- \frac{2 y \cdot 3}{15} - \frac{y}{3} \cdot \frac{5}{5} + \frac{2}{3} = - 3$

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.4.3

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

$- \frac{2 y \cdot 3}{15} - \frac{y \cdot 5}{3 \cdot 5} + \frac{2}{3} = - 3$

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.4.4

Multiply $3$ by $5$ .

$- \frac{2 y \cdot 3}{15} - \frac{y \cdot 5}{15} + \frac{2}{3} = - 3$

$x = - \frac{12 y}{5} + 4$

$- \frac{2 y \cdot 3}{15} - \frac{y \cdot 5}{15} + \frac{2}{3} = - 3$

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{- 2 y \cdot 3 - y \cdot 5}{15} + \frac{2}{3} = - 3$

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.6

Simplify each term.

Tap for more steps...

Step 2.2.1.6.1

Simplify the numerator.

Tap for more steps...

Step 2.2.1.6.1.1

Factor $y$ out of $- 2 y \cdot 3 - y \cdot 5$ .

Tap for more steps...

Step 2.2.1.6.1.1.1

Factor $y$ out of $- 2 y \cdot 3$ .

$\frac{y (- 2 \cdot 3) - y \cdot 5}{15} + \frac{2}{3} = - 3$

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.6.1.1.2

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

$\frac{y (- 2 \cdot 3) + y (- 1 \cdot 5)}{15} + \frac{2}{3} = - 3$

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.6.1.1.3

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

$\frac{y (- 2 \cdot 3 - 1 \cdot 5)}{15} + \frac{2}{3} = - 3$

$x = - \frac{12 y}{5} + 4$

$\frac{y (- 2 \cdot 3 - 1 \cdot 5)}{15} + \frac{2}{3} = - 3$

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.6.1.2

Multiply $- 2$ by $3$ .

$\frac{y (- 6 - 1 \cdot 5)}{15} + \frac{2}{3} = - 3$

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.6.1.3

Multiply $- 1$ by $5$ .

$\frac{y (- 6 - 5)}{15} + \frac{2}{3} = - 3$

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.6.1.4

Subtract $5$ from $- 6$ .

$\frac{y \cdot - 11}{15} + \frac{2}{3} = - 3$

$x = - \frac{12 y}{5} + 4$

$\frac{y \cdot - 11}{15} + \frac{2}{3} = - 3$

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.6.2

Move $- 11$ to the left of $y$ .

$\frac{- 11 \cdot y}{15} + \frac{2}{3} = - 3$

$x = - \frac{12 y}{5} + 4$

Step 2.2.1.6.3

Move the negative in front of the fraction.

$- \frac{11 y}{15} + \frac{2}{3} = - 3$

$x = - \frac{12 y}{5} + 4$

$- \frac{11 y}{15} + \frac{2}{3} = - 3$

$x = - \frac{12 y}{5} + 4$

$- \frac{11 y}{15} + \frac{2}{3} = - 3$

$x = - \frac{12 y}{5} + 4$

$- \frac{11 y}{15} + \frac{2}{3} = - 3$

$x = - \frac{12 y}{5} + 4$

$- \frac{11 y}{15} + \frac{2}{3} = - 3$

$x = - \frac{12 y}{5} + 4$

Step 3

Solve for $y$ in $- \frac{11 y}{15} + \frac{2}{3} = - 3$ .

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

Subtract $\frac{2}{3}$ from both sides of the equation.

$- \frac{11 y}{15} = - 3 - \frac{2}{3}$

$x = - \frac{12 y}{5} + 4$

Step 3.1.2

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

$- \frac{11 y}{15} = - 3 \cdot \frac{3}{3} - \frac{2}{3}$

$x = - \frac{12 y}{5} + 4$

Step 3.1.3

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

$- \frac{11 y}{15} = \frac{- 3 \cdot 3}{3} - \frac{2}{3}$

$x = - \frac{12 y}{5} + 4$

Step 3.1.4

Combine the numerators over the common denominator.

$- \frac{11 y}{15} = \frac{- 3 \cdot 3 - 2}{3}$

$x = - \frac{12 y}{5} + 4$

Step 3.1.5

Simplify the numerator.

Tap for more steps...

Step 3.1.5.1

Multiply $- 3$ by $3$ .

$- \frac{11 y}{15} = \frac{- 9 - 2}{3}$

$x = - \frac{12 y}{5} + 4$

Step 3.1.5.2

Subtract $2$ from $- 9$ .

$- \frac{11 y}{15} = \frac{- 11}{3}$

$x = - \frac{12 y}{5} + 4$

$- \frac{11 y}{15} = \frac{- 11}{3}$

$x = - \frac{12 y}{5} + 4$

Step 3.1.6

Move the negative in front of the fraction.

$- \frac{11 y}{15} = - \frac{11}{3}$

$x = - \frac{12 y}{5} + 4$

$- \frac{11 y}{15} = - \frac{11}{3}$

$x = - \frac{12 y}{5} + 4$

Step 3.2

Multiply both sides of the equation by $- \frac{15}{11}$ .

$- \frac{15}{11} \cdot (- \frac{11 y}{15}) = - \frac{15}{11} \cdot (- \frac{11}{3})$

$x = - \frac{12 y}{5} + 4$

Step 3.3

Simplify both sides of the equation.

Tap for more steps...

Step 3.3.1

Simplify the left side.

Tap for more steps...

Step 3.3.1.1

Simplify $- \frac{15}{11} (- \frac{11 y}{15})$ .

Tap for more steps...

Step 3.3.1.1.1

Cancel the common factor of $15$ .

Tap for more steps...

Step 3.3.1.1.1.1

Move the leading negative in $- \frac{15}{11}$ into the numerator.

$\frac{- 15}{11} \cdot (- \frac{11 y}{15}) = - \frac{15}{11} \cdot (- \frac{11}{3})$

$x = - \frac{12 y}{5} + 4$

Step 3.3.1.1.1.2

Move the leading negative in $- \frac{11 y}{15}$ into the numerator.

$\frac{- 15}{11} \cdot \frac{- 11 y}{15} = - \frac{15}{11} \cdot (- \frac{11}{3})$

$x = - \frac{12 y}{5} + 4$

Step 3.3.1.1.1.3

Factor $15$ out of $- 15$ .

$\frac{15 (- 1)}{11} \cdot \frac{- 11 y}{15} = - \frac{15}{11} \cdot (- \frac{11}{3})$

$x = - \frac{12 y}{5} + 4$

Step 3.3.1.1.1.4

Cancel the common factor.

$\frac{15 \cdot - 1}{11} \cdot \frac{- 11 y}{15} = - \frac{15}{11} \cdot (- \frac{11}{3})$

$x = - \frac{12 y}{5} + 4$

Step 3.3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{11} \cdot (- 11 y) = - \frac{15}{11} \cdot (- \frac{11}{3})$

$x = - \frac{12 y}{5} + 4$

$\frac{- 1}{11} \cdot (- 11 y) = - \frac{15}{11} \cdot (- \frac{11}{3})$

$x = - \frac{12 y}{5} + 4$

Step 3.3.1.1.2

Cancel the common factor of $11$ .

Tap for more steps...

Step 3.3.1.1.2.1

Factor $11$ out of $- 11 y$ .

$\frac{- 1}{11} \cdot (11 (- y)) = - \frac{15}{11} \cdot (- \frac{11}{3})$

$x = - \frac{12 y}{5} + 4$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{11} \cdot (11 (- y)) = - \frac{15}{11} \cdot (- \frac{11}{3})$

$x = - \frac{12 y}{5} + 4$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = - \frac{15}{11} \cdot (- \frac{11}{3})$

$x = - \frac{12 y}{5} + 4$

$y = - \frac{15}{11} \cdot (- \frac{11}{3})$

$x = - \frac{12 y}{5} + 4$

Step 3.3.1.1.3

Multiply.

Tap for more steps...

Step 3.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{15}{11} \cdot (- \frac{11}{3})$

$x = - \frac{12 y}{5} + 4$

Step 3.3.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{15}{11} \cdot (- \frac{11}{3})$

$x = - \frac{12 y}{5} + 4$

$y = - \frac{15}{11} \cdot (- \frac{11}{3})$

$x = - \frac{12 y}{5} + 4$

$y = - \frac{15}{11} \cdot (- \frac{11}{3})$

$x = - \frac{12 y}{5} + 4$

$y = - \frac{15}{11} \cdot (- \frac{11}{3})$

$x = - \frac{12 y}{5} + 4$

Step 3.3.2

Simplify the right side.

Tap for more steps...

Step 3.3.2.1

Simplify $- \frac{15}{11} (- \frac{11}{3})$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.3.2.1.1.1

Move the leading negative in $- \frac{15}{11}$ into the numerator.

$y = \frac{- 15}{11} \cdot (- \frac{11}{3})$

$x = - \frac{12 y}{5} + 4$

Step 3.3.2.1.1.2

Move the leading negative in $- \frac{11}{3}$ into the numerator.

$y = \frac{- 15}{11} \cdot \frac{- 11}{3}$

$x = - \frac{12 y}{5} + 4$

Step 3.3.2.1.1.3

Factor $3$ out of $- 15$ .

$y = \frac{3 (- 5)}{11} \cdot \frac{- 11}{3}$

$x = - \frac{12 y}{5} + 4$

Step 3.3.2.1.1.4

Cancel the common factor.

$y = \frac{3 \cdot - 5}{11} \cdot \frac{- 11}{3}$

$x = - \frac{12 y}{5} + 4$

Step 3.3.2.1.1.5

Rewrite the expression.

$y = \frac{- 5}{11} \cdot - 11$

$x = - \frac{12 y}{5} + 4$

$y = \frac{- 5}{11} \cdot - 11$

$x = - \frac{12 y}{5} + 4$

Step 3.3.2.1.2

Cancel the common factor of $11$ .

Tap for more steps...

Step 3.3.2.1.2.1

Factor $11$ out of $- 11$ .

$y = \frac{- 5}{11} \cdot (11 (- 1))$

$x = - \frac{12 y}{5} + 4$

Step 3.3.2.1.2.2

Cancel the common factor.

$y = \frac{- 5}{11} \cdot (11 \cdot - 1)$

$x = - \frac{12 y}{5} + 4$

Step 3.3.2.1.2.3

Rewrite the expression.

$y = - 5 \cdot - 1$

$x = - \frac{12 y}{5} + 4$

$y = - 5 \cdot - 1$

$x = - \frac{12 y}{5} + 4$

Step 3.3.2.1.3

Multiply $- 5$ by $- 1$ .

$y = 5$

$x = - \frac{12 y}{5} + 4$

$y = 5$

$x = - \frac{12 y}{5} + 4$

$y = 5$

$x = - \frac{12 y}{5} + 4$

$y = 5$

$x = - \frac{12 y}{5} + 4$

$y = 5$

$x = - \frac{12 y}{5} + 4$

Step 4

Replace all occurrences of $y$ with $5$ in each equation.

Tap for more steps...

Step 4.1

Replace all occurrences of $y$ in $x = - \frac{12 y}{5} + 4$ with $5$ .

$x = - \frac{12 (5)}{5} + 4$

$y = 5$

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Simplify $- \frac{12 (5)}{5} + 4$ .

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 $5$ .

Tap for more steps...

Step 4.2.1.1.1.1

Cancel the common factor.

$x = - \frac{12 \cdot 5}{5} + 4$

$y = 5$

Step 4.2.1.1.1.2

Divide $12$ by $1$ .

$x = - 1 \cdot 12 + 4$

$y = 5$

$x = - 1 \cdot 12 + 4$

$y = 5$

Step 4.2.1.1.2

Multiply $- 1$ by $12$ .

$x = - 12 + 4$

$y = 5$

$x = - 12 + 4$

$y = 5$

Step 4.2.1.2

Add $- 12$ and $4$ .

$x = - 8$

$y = 5$

$x = - 8$

$y = 5$

$x = - 8$

$y = 5$

$x = - 8$

$y = 5$

Step 5

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

$(- 8, 5)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 8, 5)$

Equation Form:

$x = - 8, y = 5$

Step 7