Algebra Examples

$3 x + y = \frac{1}{3}$ $2 x - 3 y = \frac{8}{3}$

Step 1

Subtract $3 x$ from both sides of the equation.

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

$2 x - 3 y = \frac{8}{3}$

Step 2

Replace all occurrences of $y$ with $\frac{1}{3} - 3 x$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $2 x - 3 y = \frac{8}{3}$ with $\frac{1}{3} - 3 x$ .

$2 x - 3 (\frac{1}{3} - 3 x) = \frac{8}{3}$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 x - 3 (\frac{1}{3} - 3 x)$ .

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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.

$2 x - 3 (\frac{1}{3}) - 3 (- 3 x) = \frac{8}{3}$

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

Step 2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1

Factor $3$ out of $- 3$ .

$2 x + 3 (- 1) (\frac{1}{3}) - 3 (- 3 x) = \frac{8}{3}$

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

Step 2.2.1.1.2.2

Cancel the common factor.

$2 x + 3 \cdot (- 1 (\frac{1}{3})) - 3 (- 3 x) = \frac{8}{3}$

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

Step 2.2.1.1.2.3

Rewrite the expression.

$2 x - 1 - 3 (- 3 x) = \frac{8}{3}$

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

$2 x - 1 - 3 (- 3 x) = \frac{8}{3}$

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

Step 2.2.1.1.3

Multiply $- 3$ by $- 3$ .

$2 x - 1 + 9 x = \frac{8}{3}$

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

$2 x - 1 + 9 x = \frac{8}{3}$

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

Step 2.2.1.2

Add $2 x$ and $9 x$ .

$11 x - 1 = \frac{8}{3}$

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

$11 x - 1 = \frac{8}{3}$

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

$11 x - 1 = \frac{8}{3}$

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

$11 x - 1 = \frac{8}{3}$

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

Step 3

Solve for $x$ in $11 x - 1 = \frac{8}{3}$ .

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Step 3.1

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

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Step 3.1.1

Add $1$ to both sides of the equation.

$11 x = \frac{8}{3} + 1$

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

Step 3.1.2

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

$11 x = \frac{8}{3} + \frac{3}{3}$

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

Step 3.1.3

Combine the numerators over the common denominator.

$11 x = \frac{8 + 3}{3}$

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

Step 3.1.4

Add $8$ and $3$ .

$11 x = \frac{11}{3}$

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

$11 x = \frac{11}{3}$

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

Step 3.2

Divide each term in $11 x = \frac{11}{3}$ by $11$ and simplify.

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Step 3.2.1

Divide each term in $11 x = \frac{11}{3}$ by $11$ .

$\frac{11 x}{11} = \frac{\frac{11}{3}}{11}$

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $11$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{11 x}{11} = \frac{\frac{11}{3}}{11}$

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

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{11}{3}}{11}$

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

$x = \frac{\frac{11}{3}}{11}$

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

$x = \frac{\frac{11}{3}}{11}$

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{11}{3} \cdot \frac{1}{11}$

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

Step 3.2.3.2

Cancel the common factor of $11$ .

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Step 3.2.3.2.1

Cancel the common factor.

$x = \frac{11}{3} \cdot \frac{1}{11}$

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

Step 3.2.3.2.2

Rewrite the expression.

$x = \frac{1}{3}$

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

$x = \frac{1}{3}$

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

$x = \frac{1}{3}$

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

$x = \frac{1}{3}$

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

$x = \frac{1}{3}$

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

Step 4

Replace all occurrences of $x$ with $\frac{1}{3}$ in each equation.

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Step 4.1

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

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

$x = \frac{1}{3}$

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

Cancel the common factor of $3$ .

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Step 4.2.1.1.1

Factor $3$ out of $- 3$ .

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

$x = \frac{1}{3}$

Step 4.2.1.1.2

Cancel the common factor.

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

$x = \frac{1}{3}$

Step 4.2.1.1.3

Rewrite the expression.

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

$x = \frac{1}{3}$

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

$x = \frac{1}{3}$

Step 4.2.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

$x = \frac{1}{3}$

Step 4.2.1.3

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

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

$x = \frac{1}{3}$

Step 4.2.1.4

Combine the numerators over the common denominator.

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

$x = \frac{1}{3}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $- 1$ by $3$ .

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

$x = \frac{1}{3}$

Step 4.2.1.5.2

Subtract $3$ from $1$ .

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

$x = \frac{1}{3}$

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

$x = \frac{1}{3}$

Step 4.2.1.6

Move the negative in front of the fraction.

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

$x = \frac{1}{3}$

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

$x = \frac{1}{3}$

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

$x = \frac{1}{3}$

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

$x = \frac{1}{3}$

Step 5

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

$(\frac{1}{3}, - \frac{2}{3})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{1}{3}, - \frac{2}{3})$

Equation Form:

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

Step 7