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

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

Step 1

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

Simplify each term.

Combine $\frac{1}{5}$ and $x$ .

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

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

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

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

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

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

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

Add $\frac{y}{4}$ to both sides of the equation.

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

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

Multiply both sides of the equation by $5$ .

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

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

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

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

Rewrite the expression.

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

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

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

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

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

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

Simplify the right side.

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

Apply the distributive property.

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

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

Multiply $5$ by $3$ .

$x = 15 + 5 (\frac{y}{4})$

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

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

$x = 15 + \frac{5 y}{4}$

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

$x = 15 + \frac{5 y}{4}$

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

$x = 15 + \frac{5 y}{4}$

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

$x = 15 + \frac{5 y}{4}$

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

Reorder $15$ and $\frac{5 y}{4}$ .

$x = \frac{5 y}{4} + 15$

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

$x = \frac{5 y}{4} + 15$

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

Step 2

Replace all occurrences of $x$ with $\frac{5 y}{4} + 15$ in each equation.

Replace all occurrences of $x$ in $\frac{2}{5} x + \frac{1}{2} y = 3$ with $\frac{5 y}{4} + 15$ .

$\frac{2}{5} \cdot (\frac{5 y}{4} + 15) + \frac{1}{2} y = 3$

$x = \frac{5 y}{4} + 15$

Simplify the left side.

Simplify $\frac{2}{5} \cdot (\frac{5 y}{4} + 15) + \frac{1}{2} y$ .

Simplify each term.

Apply the distributive property.

$\frac{2}{5} \cdot \frac{5 y}{4} + \frac{2}{5} \cdot 15 + \frac{1}{2} y = 3$

$x = \frac{5 y}{4} + 15$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

$x = \frac{5 y}{4} + 15$

Cancel the common factor.

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

$x = \frac{5 y}{4} + 15$

Rewrite the expression.

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

$x = \frac{5 y}{4} + 15$

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

$x = \frac{5 y}{4} + 15$

Cancel the common factor of $5$ .

Factor $5$ out of $5 y$ .

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

$x = \frac{5 y}{4} + 15$

Cancel the common factor.

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

$x = \frac{5 y}{4} + 15$

Rewrite the expression.

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

$x = \frac{5 y}{4} + 15$

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

$x = \frac{5 y}{4} + 15$

Cancel the common factor of $5$ .

Factor $5$ out of $15$ .

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

$x = \frac{5 y}{4} + 15$

Cancel the common factor.

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

$x = \frac{5 y}{4} + 15$

Rewrite the expression.

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

$x = \frac{5 y}{4} + 15$

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

$x = \frac{5 y}{4} + 15$

Multiply $2$ by $3$ .

$\frac{y}{2} + 6 + \frac{1}{2} y = 3$

$x = \frac{5 y}{4} + 15$

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

$\frac{y}{2} + 6 + \frac{y}{2} = 3$

$x = \frac{5 y}{4} + 15$

$\frac{y}{2} + 6 + \frac{y}{2} = 3$

$x = \frac{5 y}{4} + 15$

Simplify terms.

Combine the numerators over the common denominator.

$6 + \frac{y + y}{2} = 3$

$x = \frac{5 y}{4} + 15$

Add $y$ and $y$ .

$6 + \frac{2 y}{2} = 3$

$x = \frac{5 y}{4} + 15$

Cancel the common factor of $2$ .

Cancel the common factor.

$6 + \frac{2 y}{2} = 3$

$x = \frac{5 y}{4} + 15$

Divide $y$ by $1$ .

$6 + y = 3$

$x = \frac{5 y}{4} + 15$

$6 + y = 3$

$x = \frac{5 y}{4} + 15$

$6 + y = 3$

$x = \frac{5 y}{4} + 15$

$6 + y = 3$

$x = \frac{5 y}{4} + 15$

$6 + y = 3$

$x = \frac{5 y}{4} + 15$

$6 + y = 3$

$x = \frac{5 y}{4} + 15$

Step 3

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

Subtract $6$ from both sides of the equation.

$y = 3 - 6$

$x = \frac{5 y}{4} + 15$

Subtract $6$ from $3$ .

$y = - 3$

$x = \frac{5 y}{4} + 15$

$y = - 3$

$x = \frac{5 y}{4} + 15$

Step 4

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

Replace all occurrences of $y$ in $x = \frac{5 y}{4} + 15$ with $- 3$ .

$x = \frac{5 (- 3)}{4} + 15$

$y = - 3$

Simplify the right side.

Simplify $\frac{5 (- 3)}{4} + 15$ .

Multiply $5$ by $- 3$ .

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

$y = - 3$

Move the negative in front of the fraction.

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

$y = - 3$

To write $15$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = - \frac{15}{4} + 15 \cdot \frac{4}{4}$

$y = - 3$

Combine $15$ and $\frac{4}{4}$ .

$x = - \frac{15}{4} + \frac{15 \cdot 4}{4}$

$y = - 3$

Combine the numerators over the common denominator.

$x = \frac{- 15 + 15 \cdot 4}{4}$

$y = - 3$

Simplify the numerator.

Multiply $15$ by $4$ .

$x = \frac{- 15 + 60}{4}$

$y = - 3$

Add $- 15$ and $60$ .

$x = \frac{45}{4}$

$y = - 3$

$x = \frac{45}{4}$

$y = - 3$

$x = \frac{45}{4}$

$y = - 3$

$x = \frac{45}{4}$

$y = - 3$

$x = \frac{45}{4}$

$y = - 3$

Step 5

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

$(\frac{45}{4}, - 3)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{45}{4}, - 3)$

Equation Form:

$x = \frac{45}{4}, y = - 3$

Step 7

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