Solve the System of Equations 1/5x-2y=-2/5 1/3x+1/2y=7

$\frac{1}{5} x - 2 y = - \frac{2}{5}$ $\frac{1}{3} x + \frac{1}{2} y = 7$

Step 1

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

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

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

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

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

Divide each term in $- 2 y = - \frac{2}{5} - \frac{x}{5}$ by $- 2$ and simplify.

Divide each term in $- 2 y = - \frac{2}{5} - \frac{x}{5}$ by $- 2$ .

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

Divide $y$ by $1$ .

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

Simplify the right side.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{2}{5}$ into the numerator.

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

Factor $2$ out of $- 2$ .

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

Factor $2$ out of $- 2$ .

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

Cancel the common factor.

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

Rewrite the expression.

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

Multiply $\frac{- 1}{5}$ by $\frac{1}{- 1}$ .

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

Multiply $5$ by $- 1$ .

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

Dividing two negative values results in a positive value.

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

Multiply the numerator by the reciprocal of the denominator.

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

Move the negative in front of the fraction.

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

Multiply $- \frac{x}{5} (- \frac{1}{2})$ .

Multiply $- 1$ by $- 1$ .

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

Multiply $\frac{x}{5}$ by $1$ .

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

Multiply $\frac{x}{5}$ by $\frac{1}{2}$ .

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

$\frac{1}{3} x + \frac{1}{2} y = 7$

Multiply $5$ by $2$ .

$y = \frac{1}{5} + \frac{x}{10}$

$\frac{1}{3} x + \frac{1}{2} y = 7$

$y = \frac{1}{5} + \frac{x}{10}$

$\frac{1}{3} x + \frac{1}{2} y = 7$

$y = \frac{1}{5} + \frac{x}{10}$

$\frac{1}{3} x + \frac{1}{2} y = 7$

$y = \frac{1}{5} + \frac{x}{10}$

$\frac{1}{3} x + \frac{1}{2} y = 7$

$y = \frac{1}{5} + \frac{x}{10}$

$\frac{1}{3} x + \frac{1}{2} y = 7$

$y = \frac{1}{5} + \frac{x}{10}$

$\frac{1}{3} x + \frac{1}{2} y = 7$

Step 2

Replace all occurrences of $y$ with $\frac{1}{5} + \frac{x}{10}$ in each equation.

Replace all occurrences of $y$ in $\frac{1}{3} x + \frac{1}{2} y = 7$ with $\frac{1}{5} + \frac{x}{10}$ .

$\frac{1}{3} x + \frac{1}{2} \cdot (\frac{1}{5} + \frac{x}{10}) = 7$

$y = \frac{1}{5} + \frac{x}{10}$

Simplify the left side.

Simplify $\frac{1}{3} x + \frac{1}{2} \cdot (\frac{1}{5} + \frac{x}{10})$ .

Simplify each term.

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

$\frac{x}{3} + \frac{1}{2} \cdot (\frac{1}{5} + \frac{x}{10}) = 7$

$y = \frac{1}{5} + \frac{x}{10}$

Apply the distributive property.

$\frac{x}{3} + \frac{1}{2} \cdot \frac{1}{5} + \frac{1}{2} \cdot \frac{x}{10} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

Multiply $\frac{1}{2} \cdot \frac{1}{5}$ .

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

$\frac{x}{3} + \frac{1}{2 \cdot 5} + \frac{1}{2} \cdot \frac{x}{10} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

Multiply $2$ by $5$ .

$\frac{x}{3} + \frac{1}{10} + \frac{1}{2} \cdot \frac{x}{10} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

$\frac{x}{3} + \frac{1}{10} + \frac{1}{2} \cdot \frac{x}{10} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

Multiply $\frac{1}{2} \cdot \frac{x}{10}$ .

Multiply $\frac{1}{2}$ by $\frac{x}{10}$ .

$\frac{x}{3} + \frac{1}{10} + \frac{x}{2 \cdot 10} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

Multiply $2$ by $10$ .

$\frac{x}{3} + \frac{1}{10} + \frac{x}{20} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

$\frac{x}{3} + \frac{1}{10} + \frac{x}{20} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

$\frac{x}{3} + \frac{1}{10} + \frac{x}{20} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

To write $\frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$\frac{x}{3} \cdot \frac{20}{20} + \frac{x}{20} + \frac{1}{10} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

To write $\frac{x}{20}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x}{3} \cdot \frac{20}{20} + \frac{x}{20} \cdot \frac{3}{3} + \frac{1}{10} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

Write each expression with a common denominator of $60$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x}{3}$ by $\frac{20}{20}$ .

$\frac{x \cdot 20}{3 \cdot 20} + \frac{x}{20} \cdot \frac{3}{3} + \frac{1}{10} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

Multiply $3$ by $20$ .

$\frac{x \cdot 20}{60} + \frac{x}{20} \cdot \frac{3}{3} + \frac{1}{10} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

Multiply $\frac{x}{20}$ by $\frac{3}{3}$ .

$\frac{x \cdot 20}{60} + \frac{x \cdot 3}{20 \cdot 3} + \frac{1}{10} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

Multiply $20$ by $3$ .

$\frac{x \cdot 20}{60} + \frac{x \cdot 3}{60} + \frac{1}{10} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

$\frac{x \cdot 20}{60} + \frac{x \cdot 3}{60} + \frac{1}{10} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

Combine the numerators over the common denominator.

$\frac{x \cdot 20 + x \cdot 3}{60} + \frac{1}{10} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

Reorder $x$ and $3$ .

$\frac{20 \cdot x + 3 \cdot x}{60} + \frac{1}{10} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

Add $20 \cdot x$ and $3 \cdot x$ .

$\frac{23 x}{60} + \frac{1}{10} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

$\frac{23 x}{60} + \frac{1}{10} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

$\frac{23 x}{60} + \frac{1}{10} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

$\frac{23 x}{60} + \frac{1}{10} = 7$

$y = \frac{1}{5} + \frac{x}{10}$

Step 3

Solve for $x$ in $\frac{23 x}{60} + \frac{1}{10} = 7$ .

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

Subtract $\frac{1}{10}$ from both sides of the equation.

$\frac{23 x}{60} = 7 - \frac{1}{10}$

$y = \frac{1}{5} + \frac{x}{10}$

To write $7$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{23 x}{60} = 7 \cdot \frac{10}{10} - \frac{1}{10}$

$y = \frac{1}{5} + \frac{x}{10}$

Combine $7$ and $\frac{10}{10}$ .

$\frac{23 x}{60} = \frac{7 \cdot 10}{10} - \frac{1}{10}$

$y = \frac{1}{5} + \frac{x}{10}$

Combine the numerators over the common denominator.

$\frac{23 x}{60} = \frac{7 \cdot 10 - 1}{10}$

$y = \frac{1}{5} + \frac{x}{10}$

Simplify the numerator.

Multiply $7$ by $10$ .

$\frac{23 x}{60} = \frac{70 - 1}{10}$

$y = \frac{1}{5} + \frac{x}{10}$

Subtract $1$ from $70$ .

$\frac{23 x}{60} = \frac{69}{10}$

$y = \frac{1}{5} + \frac{x}{10}$

$\frac{23 x}{60} = \frac{69}{10}$

$y = \frac{1}{5} + \frac{x}{10}$

$\frac{23 x}{60} = \frac{69}{10}$

$y = \frac{1}{5} + \frac{x}{10}$

Multiply both sides of the equation by $\frac{60}{23}$ .

$\frac{60}{23} \cdot \frac{23 x}{60} = \frac{60}{23} \cdot \frac{69}{10}$

$y = \frac{1}{5} + \frac{x}{10}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{60}{23} \cdot \frac{23 x}{60}$ .

Cancel the common factor of $60$ .

Cancel the common factor.

$\frac{60}{23} \cdot \frac{23 x}{60} = \frac{60}{23} \cdot \frac{69}{10}$

$y = \frac{1}{5} + \frac{x}{10}$

Rewrite the expression.

$\frac{1}{23} \cdot (23 x) = \frac{60}{23} \cdot \frac{69}{10}$

$y = \frac{1}{5} + \frac{x}{10}$

$\frac{1}{23} \cdot (23 x) = \frac{60}{23} \cdot \frac{69}{10}$

$y = \frac{1}{5} + \frac{x}{10}$

Cancel the common factor of $23$ .

Factor $23$ out of $23 x$ .

$\frac{1}{23} \cdot (23 (x)) = \frac{60}{23} \cdot \frac{69}{10}$

$y = \frac{1}{5} + \frac{x}{10}$

Cancel the common factor.

$\frac{1}{23} \cdot (23 x) = \frac{60}{23} \cdot \frac{69}{10}$

$y = \frac{1}{5} + \frac{x}{10}$

Rewrite the expression.

$x = \frac{60}{23} \cdot \frac{69}{10}$

$y = \frac{1}{5} + \frac{x}{10}$

$x = \frac{60}{23} \cdot \frac{69}{10}$

$y = \frac{1}{5} + \frac{x}{10}$

$x = \frac{60}{23} \cdot \frac{69}{10}$

$y = \frac{1}{5} + \frac{x}{10}$

$x = \frac{60}{23} \cdot \frac{69}{10}$

$y = \frac{1}{5} + \frac{x}{10}$

Simplify the right side.

Simplify $\frac{60}{23} \cdot \frac{69}{10}$ .

Cancel the common factor of $10$ .

Factor $10$ out of $60$ .

$x = \frac{10 (6)}{23} \cdot \frac{69}{10}$

$y = \frac{1}{5} + \frac{x}{10}$

Cancel the common factor.

$x = \frac{10 \cdot 6}{23} \cdot \frac{69}{10}$

$y = \frac{1}{5} + \frac{x}{10}$

Rewrite the expression.

$x = \frac{6}{23} \cdot 69$

$y = \frac{1}{5} + \frac{x}{10}$

$x = \frac{6}{23} \cdot 69$

$y = \frac{1}{5} + \frac{x}{10}$

Cancel the common factor of $23$ .

Factor $23$ out of $69$ .

$x = \frac{6}{23} \cdot (23 (3))$

$y = \frac{1}{5} + \frac{x}{10}$

Cancel the common factor.

$x = \frac{6}{23} \cdot (23 \cdot 3)$

$y = \frac{1}{5} + \frac{x}{10}$

Rewrite the expression.

$x = 6 \cdot 3$

$y = \frac{1}{5} + \frac{x}{10}$

$x = 6 \cdot 3$

$y = \frac{1}{5} + \frac{x}{10}$

Multiply $6$ by $3$ .

$x = 18$

$y = \frac{1}{5} + \frac{x}{10}$

$x = 18$

$y = \frac{1}{5} + \frac{x}{10}$

$x = 18$

$y = \frac{1}{5} + \frac{x}{10}$

$x = 18$

$y = \frac{1}{5} + \frac{x}{10}$

$x = 18$

$y = \frac{1}{5} + \frac{x}{10}$

Step 4

Replace all occurrences of $x$ with $18$ in each equation.

Replace all occurrences of $x$ in $y = \frac{1}{5} + \frac{x}{10}$ with $18$ .

$y = \frac{1}{5} + \frac{18}{10}$

$x = 18$

Simplify the right side.

Simplify $\frac{1}{5} + \frac{18}{10}$ .

Cancel the common factor of $18$ and $10$ .

Factor $2$ out of $18$ .

$y = \frac{1}{5} + \frac{2 (9)}{10}$

$x = 18$

Cancel the common factors.

Factor $2$ out of $10$ .

$y = \frac{1}{5} + \frac{2 \cdot 9}{2 \cdot 5}$

$x = 18$

Cancel the common factor.

$y = \frac{1}{5} + \frac{2 \cdot 9}{2 \cdot 5}$

$x = 18$

Rewrite the expression.

$y = \frac{1}{5} + \frac{9}{5}$

$x = 18$

$y = \frac{1}{5} + \frac{9}{5}$

$x = 18$

$y = \frac{1}{5} + \frac{9}{5}$

$x = 18$

Combine the numerators over the common denominator.

$y = \frac{1 + 9}{5}$

$x = 18$

Simplify the expression.

Add $1$ and $9$ .

$y = \frac{10}{5}$

$x = 18$

Divide $10$ by $5$ .

$y = 2$

$x = 18$

$y = 2$

$x = 18$

$y = 2$

$x = 18$

$y = 2$

$x = 18$

$y = 2$

$x = 18$

Step 5

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

$(18, 2)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(18, 2)$

Equation Form:

$x = 18, y = 2$

Step 7

Solve the System of Equations 1/5x-2y=-2/5 1/3x+1/2y=7

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