Algebra Examples

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

Step 1

Combine $\frac{2}{3}$ and $x$ .

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

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

Step 2

Replace all occurrences of $y$ with $\frac{2 x}{3} - 3$ in each equation.

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Step 2.1

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

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

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

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

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

Step 2.2.1.1.2

Apply the distributive property.

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

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

Step 2.2.1.1.3

Multiply $2 \frac{2 x}{3}$ .

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Step 2.2.1.1.3.1

Combine $2$ and $\frac{2 x}{3}$ .

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

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

Step 2.2.1.1.3.2

Multiply $2$ by $2$ .

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

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

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

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

Step 2.2.1.1.4

Multiply $2$ by $- 3$ .

$\frac{x}{3} + \frac{4 x}{3} - 6 = - 1$

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

$\frac{x}{3} + \frac{4 x}{3} - 6 = - 1$

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

Step 2.2.1.2

Simplify terms.

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Step 2.2.1.2.1

Combine the numerators over the common denominator.

$- 6 + \frac{x + 4 x}{3} = - 1$

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

Step 2.2.1.2.2

Add $x$ and $4 x$ .

$- 6 + \frac{5 x}{3} = - 1$

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

$- 6 + \frac{5 x}{3} = - 1$

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

$- 6 + \frac{5 x}{3} = - 1$

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

$- 6 + \frac{5 x}{3} = - 1$

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

$- 6 + \frac{5 x}{3} = - 1$

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

Step 3

Solve for $x$ in $- 6 + \frac{5 x}{3} = - 1$ .

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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 $6$ to both sides of the equation.

$\frac{5 x}{3} = - 1 + 6$

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

Step 3.1.2

Add $- 1$ and $6$ .

$\frac{5 x}{3} = 5$

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

$\frac{5 x}{3} = 5$

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

Step 3.2

Multiply both sides of the equation by $\frac{3}{5}$ .

$\frac{3}{5} \cdot \frac{5 x}{3} = \frac{3}{5} \cdot 5$

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

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 $\frac{3}{5} \cdot \frac{5 x}{3}$ .

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Step 3.3.1.1.1

Cancel the common factor of $3$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{3}{5} \cdot \frac{5 x}{3} = \frac{3}{5} \cdot 5$

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

Step 3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{5} \cdot (5 x) = \frac{3}{5} \cdot 5$

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

$\frac{1}{5} \cdot (5 x) = \frac{3}{5} \cdot 5$

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

Step 3.3.1.1.2

Cancel the common factor of $5$ .

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Step 3.3.1.1.2.1

Factor $5$ out of $5 x$ .

$\frac{1}{5} \cdot (5 (x)) = \frac{3}{5} \cdot 5$

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

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{1}{5} \cdot (5 x) = \frac{3}{5} \cdot 5$

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

Step 3.3.1.1.2.3

Rewrite the expression.

$x = \frac{3}{5} \cdot 5$

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

$x = \frac{3}{5} \cdot 5$

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

$x = \frac{3}{5} \cdot 5$

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

$x = \frac{3}{5} \cdot 5$

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

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Cancel the common factor of $5$ .

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Step 3.3.2.1.1

Cancel the common factor.

$x = \frac{3}{5} \cdot 5$

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

Step 3.3.2.1.2

Rewrite the expression.

$x = 3$

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

$x = 3$

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

$x = 3$

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

$x = 3$

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

$x = 3$

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

Step 4

Replace all occurrences of $x$ with $3$ in each equation.

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Step 4.1

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

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

$x = 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

Multiply $2$ by $3$ .

$y = \frac{6}{3} - 3$

$x = 3$

Step 4.2.1.2

Divide $6$ by $3$ .

$y = 2 - 3$

$x = 3$

Step 4.2.1.3

Subtract $3$ from $2$ .

$y = - 1$

$x = 3$

$y = - 1$

$x = 3$

$y = - 1$

$x = 3$

$y = - 1$

$x = 3$

Step 5

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

$(3, - 1)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(3, - 1)$

Equation Form:

$x = 3, y = - 1$

Step 7