Algebra Examples

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

Step 1

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

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Step 1.1

Add $\frac{y}{2}$ to both sides of the equation.

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

$3 x - 2 y = 1$

Step 1.2

Divide each term in $5 x = - 1 + \frac{y}{2}$ by $5$ and simplify.

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Step 1.2.1

Divide each term in $5 x = - 1 + \frac{y}{2}$ by $5$ .

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

$3 x - 2 y = 1$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $5$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

$3 x - 2 y = 1$

Step 1.2.2.1.2

Divide $x$ by $1$ .

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

$3 x - 2 y = 1$

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

$3 x - 2 y = 1$

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

$3 x - 2 y = 1$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Move the negative in front of the fraction.

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

$3 x - 2 y = 1$

Step 1.2.3.1.2

Multiply the numerator by the reciprocal of the denominator.

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

$3 x - 2 y = 1$

Step 1.2.3.1.3

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

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Step 1.2.3.1.3.1

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

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

$3 x - 2 y = 1$

Step 1.2.3.1.3.2

Multiply $2$ by $5$ .

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

$3 x - 2 y = 1$

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

$3 x - 2 y = 1$

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

$3 x - 2 y = 1$

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

$3 x - 2 y = 1$

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

$3 x - 2 y = 1$

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

$3 x - 2 y = 1$

Step 2

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

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Step 2.1

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

$3 (- \frac{1}{5} + \frac{y}{10}) - 2 y = 1$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (- \frac{1}{5} + \frac{y}{10}) - 2 y$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$3 (- \frac{1}{5}) + 3 (\frac{y}{10}) - 2 y = 1$

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

Step 2.2.1.1.2

Multiply $3 (- \frac{1}{5})$ .

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Step 2.2.1.1.2.1

Multiply $- 1$ by $3$ .

$- 3 (\frac{1}{5}) + 3 (\frac{y}{10}) - 2 y = 1$

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

Step 2.2.1.1.2.2

Combine $- 3$ and $\frac{1}{5}$ .

$\frac{- 3}{5} + 3 (\frac{y}{10}) - 2 y = 1$

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

$\frac{- 3}{5} + 3 (\frac{y}{10}) - 2 y = 1$

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

Step 2.2.1.1.3

Combine $3$ and $\frac{y}{10}$ .

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

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

Step 2.2.1.1.4

Move the negative in front of the fraction.

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

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

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

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

Step 2.2.1.2

To write $- 2 y$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

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

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

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

Combine $- 2 y$ and $\frac{10}{10}$ .

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

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

Step 2.2.1.3.2

Combine the numerators over the common denominator.

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

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

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

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

Step 2.2.1.4

Simplify each term.

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Step 2.2.1.4.1

Simplify the numerator.

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Step 2.2.1.4.1.1

Factor $y$ out of $3 y - 2 y \cdot 10$ .

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Step 2.2.1.4.1.1.1

Factor $y$ out of $3 y$ .

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

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

Step 2.2.1.4.1.1.2

Factor $y$ out of $- 2 y \cdot 10$ .

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

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

Step 2.2.1.4.1.1.3

Factor $y$ out of $y \cdot 3 + y (- 2 \cdot 10)$ .

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

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

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

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

Step 2.2.1.4.1.2

Multiply $- 2$ by $10$ .

$- \frac{3}{5} + \frac{y (3 - 20)}{10} = 1$

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

Step 2.2.1.4.1.3

Subtract $20$ from $3$ .

$- \frac{3}{5} + \frac{y \cdot - 17}{10} = 1$

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

$- \frac{3}{5} + \frac{y \cdot - 17}{10} = 1$

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

Step 2.2.1.4.2

Move $- 17$ to the left of $y$ .

$- \frac{3}{5} + \frac{- 17 \cdot y}{10} = 1$

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

Step 2.2.1.4.3

Move the negative in front of the fraction.

$- \frac{3}{5} - \frac{17 y}{10} = 1$

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

$- \frac{3}{5} - \frac{17 y}{10} = 1$

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

$- \frac{3}{5} - \frac{17 y}{10} = 1$

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

$- \frac{3}{5} - \frac{17 y}{10} = 1$

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

$- \frac{3}{5} - \frac{17 y}{10} = 1$

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

Step 3

Solve for $y$ in $- \frac{3}{5} - \frac{17 y}{10} = 1$ .

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Step 3.1

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

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Step 3.1.1

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

$- \frac{17 y}{10} = 1 + \frac{3}{5}$

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

Step 3.1.2

Write $1$ as a fraction with a common denominator.

$- \frac{17 y}{10} = \frac{5}{5} + \frac{3}{5}$

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

Step 3.1.3

Combine the numerators over the common denominator.

$- \frac{17 y}{10} = \frac{5 + 3}{5}$

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

Step 3.1.4

Add $5$ and $3$ .

$- \frac{17 y}{10} = \frac{8}{5}$

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

$- \frac{17 y}{10} = \frac{8}{5}$

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

Step 3.2

Multiply both sides of the equation by $- \frac{10}{17}$ .

$- \frac{10}{17} \cdot (- \frac{17 y}{10}) = - \frac{10}{17} \cdot \frac{8}{5}$

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

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $- \frac{10}{17} (- \frac{17 y}{10})$ .

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Step 3.3.1.1.1

Cancel the common factor of $10$ .

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Step 3.3.1.1.1.1

Move the leading negative in $- \frac{10}{17}$ into the numerator.

$\frac{- 10}{17} \cdot (- \frac{17 y}{10}) = - \frac{10}{17} \cdot \frac{8}{5}$

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

Step 3.3.1.1.1.2

Move the leading negative in $- \frac{17 y}{10}$ into the numerator.

$\frac{- 10}{17} \cdot \frac{- 17 y}{10} = - \frac{10}{17} \cdot \frac{8}{5}$

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

Step 3.3.1.1.1.3

Factor $10$ out of $- 10$ .

$\frac{10 (- 1)}{17} \cdot \frac{- 17 y}{10} = - \frac{10}{17} \cdot \frac{8}{5}$

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

Step 3.3.1.1.1.4

Cancel the common factor.

$\frac{10 \cdot - 1}{17} \cdot \frac{- 17 y}{10} = - \frac{10}{17} \cdot \frac{8}{5}$

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

Step 3.3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{17} \cdot (- 17 y) = - \frac{10}{17} \cdot \frac{8}{5}$

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

$\frac{- 1}{17} \cdot (- 17 y) = - \frac{10}{17} \cdot \frac{8}{5}$

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

Step 3.3.1.1.2

Cancel the common factor of $17$ .

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Step 3.3.1.1.2.1

Factor $17$ out of $- 17 y$ .

$\frac{- 1}{17} \cdot (17 (- y)) = - \frac{10}{17} \cdot \frac{8}{5}$

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

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{17} \cdot (17 (- y)) = - \frac{10}{17} \cdot \frac{8}{5}$

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

Step 3.3.1.1.2.3

Rewrite the expression.

$y = - \frac{10}{17} \cdot \frac{8}{5}$

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

$y = - \frac{10}{17} \cdot \frac{8}{5}$

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

Step 3.3.1.1.3

Multiply.

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Step 3.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{10}{17} \cdot \frac{8}{5}$

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

Step 3.3.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{10}{17} \cdot \frac{8}{5}$

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

$y = - \frac{10}{17} \cdot \frac{8}{5}$

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

$y = - \frac{10}{17} \cdot \frac{8}{5}$

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

$y = - \frac{10}{17} \cdot \frac{8}{5}$

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

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \frac{10}{17} \cdot \frac{8}{5}$ .

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Step 3.3.2.1.1

Cancel the common factor of $5$ .

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Step 3.3.2.1.1.1

Move the leading negative in $- \frac{10}{17}$ into the numerator.

$y = \frac{- 10}{17} \cdot \frac{8}{5}$

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

Step 3.3.2.1.1.2

Factor $5$ out of $- 10$ .

$y = \frac{5 (- 2)}{17} \cdot \frac{8}{5}$

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

Step 3.3.2.1.1.3

Cancel the common factor.

$y = \frac{5 \cdot - 2}{17} \cdot \frac{8}{5}$

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

Step 3.3.2.1.1.4

Rewrite the expression.

$y = \frac{- 2}{17} \cdot 8$

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

$y = \frac{- 2}{17} \cdot 8$

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

Step 3.3.2.1.2

Combine $\frac{- 2}{17}$ and $8$ .

$y = \frac{- 2 \cdot 8}{17}$

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

Step 3.3.2.1.3

Simplify the expression.

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Step 3.3.2.1.3.1

Multiply $- 2$ by $8$ .

$y = \frac{- 16}{17}$

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

Step 3.3.2.1.3.2

Move the negative in front of the fraction.

$y = - \frac{16}{17}$

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

$y = - \frac{16}{17}$

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

$y = - \frac{16}{17}$

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

$y = - \frac{16}{17}$

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

$y = - \frac{16}{17}$

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

$y = - \frac{16}{17}$

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

Step 4

Replace all occurrences of $y$ with $- \frac{16}{17}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{1}{5} + \frac{y}{10}$ with $- \frac{16}{17}$ .

$x = - \frac{1}{5} + \frac{- \frac{16}{17}}{10}$

$y = - \frac{16}{17}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{1}{5} + \frac{- \frac{16}{17}}{10}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{1}{5} - \frac{16}{17} \cdot \frac{1}{10}$

$y = - \frac{16}{17}$

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Move the leading negative in $- \frac{16}{17}$ into the numerator.

$x = - \frac{1}{5} + \frac{- 16}{17} \cdot \frac{1}{10}$

$y = - \frac{16}{17}$

Step 4.2.1.1.2.2

Factor $2$ out of $- 16$ .

$x = - \frac{1}{5} + \frac{2 (- 8)}{17} \cdot \frac{1}{10}$

$y = - \frac{16}{17}$

Step 4.2.1.1.2.3

Factor $2$ out of $10$ .

$x = - \frac{1}{5} + \frac{2 \cdot - 8}{17} \cdot \frac{1}{2 \cdot 5}$

$y = - \frac{16}{17}$

Step 4.2.1.1.2.4

Cancel the common factor.

$x = - \frac{1}{5} + \frac{2 \cdot - 8}{17} \cdot \frac{1}{2 \cdot 5}$

$y = - \frac{16}{17}$

Step 4.2.1.1.2.5

Rewrite the expression.

$x = - \frac{1}{5} + \frac{- 8}{17} \cdot \frac{1}{5}$

$y = - \frac{16}{17}$

$x = - \frac{1}{5} + \frac{- 8}{17} \cdot \frac{1}{5}$

$y = - \frac{16}{17}$

Step 4.2.1.1.3

Multiply $\frac{- 8}{17}$ by $\frac{1}{5}$ .

$x = - \frac{1}{5} + \frac{- 8}{17 \cdot 5}$

$y = - \frac{16}{17}$

Step 4.2.1.1.4

Multiply $17$ by $5$ .

$x = - \frac{1}{5} + \frac{- 8}{85}$

$y = - \frac{16}{17}$

Step 4.2.1.1.5

Move the negative in front of the fraction.

$x = - \frac{1}{5} - \frac{8}{85}$

$y = - \frac{16}{17}$

$x = - \frac{1}{5} - \frac{8}{85}$

$y = - \frac{16}{17}$

Step 4.2.1.2

To write $- \frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{17}{17}$ .

$x = - \frac{1}{5} \cdot \frac{17}{17} - \frac{8}{85}$

$y = - \frac{16}{17}$

Step 4.2.1.3

Write each expression with a common denominator of $85$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1.3.1

Multiply $\frac{1}{5}$ by $\frac{17}{17}$ .

$x = - \frac{17}{5 \cdot 17} - \frac{8}{85}$

$y = - \frac{16}{17}$

Step 4.2.1.3.2

Multiply $5$ by $17$ .

$x = - \frac{17}{85} - \frac{8}{85}$

$y = - \frac{16}{17}$

$x = - \frac{17}{85} - \frac{8}{85}$

$y = - \frac{16}{17}$

Step 4.2.1.4

Simplify the expression.

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Step 4.2.1.4.1

Combine the numerators over the common denominator.

$x = \frac{- 17 - 8}{85}$

$y = - \frac{16}{17}$

Step 4.2.1.4.2

Subtract $8$ from $- 17$ .

$x = \frac{- 25}{85}$

$y = - \frac{16}{17}$

$x = \frac{- 25}{85}$

$y = - \frac{16}{17}$

Step 4.2.1.5

Cancel the common factor of $- 25$ and $85$ .

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Step 4.2.1.5.1

Factor $5$ out of $- 25$ .

$x = \frac{5 (- 5)}{85}$

$y = - \frac{16}{17}$

Step 4.2.1.5.2

Cancel the common factors.

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Step 4.2.1.5.2.1

Factor $5$ out of $85$ .

$x = \frac{5 \cdot - 5}{5 \cdot 17}$

$y = - \frac{16}{17}$

Step 4.2.1.5.2.2

Cancel the common factor.

$x = \frac{5 \cdot - 5}{5 \cdot 17}$

$y = - \frac{16}{17}$

Step 4.2.1.5.2.3

Rewrite the expression.

$x = \frac{- 5}{17}$

$y = - \frac{16}{17}$

$x = \frac{- 5}{17}$

$y = - \frac{16}{17}$

$x = \frac{- 5}{17}$

$y = - \frac{16}{17}$

Step 4.2.1.6

Move the negative in front of the fraction.

$x = - \frac{5}{17}$

$y = - \frac{16}{17}$

$x = - \frac{5}{17}$

$y = - \frac{16}{17}$

$x = - \frac{5}{17}$

$y = - \frac{16}{17}$

$x = - \frac{5}{17}$

$y = - \frac{16}{17}$

Step 5

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

$(- \frac{5}{17}, - \frac{16}{17})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- \frac{5}{17}, - \frac{16}{17})$

Equation Form:

$x = - \frac{5}{17}, y = - \frac{16}{17}$

Step 7