Solve the System of Equations 1/2x-3/5y=-2 1/4x-1/2y=3

$\frac{1}{2} x - \frac{3}{5} y = - 2$ $\frac{1}{4} x - \frac{1}{2} y = 3$

Step 1

Solve for $x$ in $\frac{1}{2} x - \frac{3}{5} y = - 2$ .

Simplify each term.

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

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

$\frac{1}{4} x - \frac{1}{2} y = 3$

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

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

$\frac{1}{4} x - \frac{1}{2} y = 3$

Move $3$ to the left of $y$ .

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

$\frac{1}{4} x - \frac{1}{2} y = 3$

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

$\frac{1}{4} x - \frac{1}{2} y = 3$

Add $\frac{3 y}{5}$ to both sides of the equation.

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

$\frac{1}{4} x - \frac{1}{2} y = 3$

Multiply both sides of the equation by $2$ .

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

$\frac{1}{4} x - \frac{1}{2} y = 3$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

$\frac{1}{4} x - \frac{1}{2} y = 3$

Rewrite the expression.

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

$\frac{1}{4} x - \frac{1}{2} y = 3$

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

$\frac{1}{4} x - \frac{1}{2} y = 3$

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

$\frac{1}{4} x - \frac{1}{2} y = 3$

Simplify the right side.

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

Apply the distributive property.

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

$\frac{1}{4} x - \frac{1}{2} y = 3$

Multiply $2$ by $- 2$ .

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

$\frac{1}{4} x - \frac{1}{2} y = 3$

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

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

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

$\frac{1}{4} x - \frac{1}{2} y = 3$

Multiply $3$ by $2$ .

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

$\frac{1}{4} x - \frac{1}{2} y = 3$

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

$\frac{1}{4} x - \frac{1}{2} y = 3$

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

$\frac{1}{4} x - \frac{1}{2} y = 3$

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

$\frac{1}{4} x - \frac{1}{2} y = 3$

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

$\frac{1}{4} x - \frac{1}{2} y = 3$

Reorder $- 4$ and $\frac{6 y}{5}$ .

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

$\frac{1}{4} x - \frac{1}{2} y = 3$

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

$\frac{1}{4} x - \frac{1}{2} y = 3$

Step 2

Replace all occurrences of $x$ with $\frac{6 y}{5} - 4$ in each equation.

Replace all occurrences of $x$ in $\frac{1}{4} x - \frac{1}{2} y = 3$ with $\frac{6 y}{5} - 4$ .

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

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

Simplify the left side.

Simplify $\frac{1}{4} \cdot (\frac{6 y}{5} - 4) - \frac{1}{2} y$ .

Simplify each term.

Apply the distributive property.

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

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

Factor $2$ out of $6 y$ .

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

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

Cancel the common factor.

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

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

Rewrite the expression.

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

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

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

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

Multiply $\frac{1}{2}$ by $\frac{3 y}{5}$ .

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

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

Multiply $2$ by $5$ .

$\frac{3 y}{10} + \frac{1}{4} \cdot - 4 - \frac{1}{2} y = 3$

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

Cancel the common factor of $4$ .

Factor $4$ out of $- 4$ .

$\frac{3 y}{10} + \frac{1}{4} \cdot (4 (- 1)) - \frac{1}{2} y = 3$

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

Cancel the common factor.

$\frac{3 y}{10} + \frac{1}{4} \cdot (4 \cdot - 1) - \frac{1}{2} y = 3$

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

Rewrite the expression.

$\frac{3 y}{10} - 1 - \frac{1}{2} y = 3$

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

$\frac{3 y}{10} - 1 - \frac{1}{2} y = 3$

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

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

$\frac{3 y}{10} - 1 - \frac{y}{2} = 3$

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

$\frac{3 y}{10} - 1 - \frac{y}{2} = 3$

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

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

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

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

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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

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

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

Multiply $2$ by $5$ .

$\frac{3 y}{10} - \frac{y \cdot 5}{10} - 1 = 3$

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

$\frac{3 y}{10} - \frac{y \cdot 5}{10} - 1 = 3$

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

Combine the numerators over the common denominator.

$\frac{3 y - y \cdot 5}{10} - 1 = 3$

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

Simplify each term.

Simplify the numerator.

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

Factor $y$ out of $3 y$ .

$\frac{y \cdot 3 - y \cdot 5}{10} - 1 = 3$

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

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

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

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

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

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

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

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

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

Multiply $- 1$ by $5$ .

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

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

Subtract $5$ from $3$ .

$\frac{y \cdot - 2}{10} - 1 = 3$

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

$\frac{y \cdot - 2}{10} - 1 = 3$

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

Cancel the common factor of $- 2$ and $10$ .

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

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

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

Cancel the common factors.

Factor $2$ out of $10$ .

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

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

Cancel the common factor.

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

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

Rewrite the expression.

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

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

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

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

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

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

Move $- 1$ to the left of $y$ .

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

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

Move the negative in front of the fraction.

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

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

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

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

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

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

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

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

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

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

Step 3

Solve for $y$ in $- \frac{y}{5} - 1 = 3$ .

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

Add $1$ to both sides of the equation.

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

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

Add $3$ and $1$ .

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

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

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

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

Multiply both sides of the equation by $- 5$ .

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

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

Simplify both sides of the equation.

Simplify the left side.

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

Cancel the common factor of $5$ .

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

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

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

Factor $5$ out of $- 5$ .

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

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

Cancel the common factor.

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

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

Rewrite the expression.

$y = - 5 \cdot 4$

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

$y = - 5 \cdot 4$

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

Multiply.

Multiply $- 1$ by $- 1$ .

$1 y = - 5 \cdot 4$

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

Multiply $y$ by $1$ .

$y = - 5 \cdot 4$

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

$y = - 5 \cdot 4$

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

$y = - 5 \cdot 4$

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

$y = - 5 \cdot 4$

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

Simplify the right side.

Multiply $- 5$ by $4$ .

$y = - 20$

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

$y = - 20$

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

$y = - 20$

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

$y = - 20$

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

Step 4

Replace all occurrences of $y$ with $- 20$ in each equation.

Replace all occurrences of $y$ in $x = \frac{6 y}{5} - 4$ with $- 20$ .

$x = \frac{6 (- 20)}{5} - 4$

$y = - 20$

Simplify the right side.

Simplify $\frac{6 (- 20)}{5} - 4$ .

Multiply $6$ by $- 20$ .

$x = \frac{- 120}{5} - 4$

$y = - 20$

Divide $- 120$ by $5$ .

$x = - 24 - 4$

$y = - 20$

Subtract $4$ from $- 24$ .

$x = - 28$

$y = - 20$

$x = - 28$

$y = - 20$

$x = - 28$

$y = - 20$

$x = - 28$

$y = - 20$

Step 5

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

$(- 28, - 20)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 28, - 20)$

Equation Form:

$x = - 28, y = - 20$

Step 7

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