Algebra Examples

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

Step 1

Eliminate the equal sides of each equation and combine.

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

Step 2

Solve $\frac{1}{3} x + 2 = \frac{2}{3} x + 5$ for $x$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Move all terms containing $x$ to the left side of the equation.

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Step 2.3.1

Subtract $\frac{2 x}{3}$ from both sides of the equation.

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

Step 2.3.2

Combine the numerators over the common denominator.

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

Step 2.3.3

Subtract $2 x$ from $x$ .

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

Step 2.3.4

Move the negative in front of the fraction.

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

Step 2.4

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

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Step 2.4.1

Subtract $2$ from both sides of the equation.

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

Step 2.4.2

Subtract $2$ from $5$ .

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

Step 2.5

Multiply both sides of the equation by $- 3$ .

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

Step 2.6

Simplify both sides of the equation.

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Step 2.6.1

Simplify the left side.

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Step 2.6.1.1

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

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Step 2.6.1.1.1

Cancel the common factor of $3$ .

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Step 2.6.1.1.1.1

Move the leading negative in $- \frac{x}{3}$ into the numerator.

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

Step 2.6.1.1.1.2

Factor $3$ out of $- 3$ .

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

Step 2.6.1.1.1.3

Cancel the common factor.

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

Step 2.6.1.1.1.4

Rewrite the expression.

$- - x = - 3 \cdot 3$

Step 2.6.1.1.2

Multiply.

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Step 2.6.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 x = - 3 \cdot 3$

Step 2.6.1.1.2.2

Multiply $x$ by $1$ .

$x = - 3 \cdot 3$

Step 2.6.2

Simplify the right side.

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Step 2.6.2.1

Multiply $- 3$ by $3$ .

$x = - 9$

Step 3

Evaluate $y$ when $x = - 9$ .

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Step 3.1

Substitute $- 9$ for $x$ .

$y = \frac{2}{3} \cdot (- 9) + 5$

Step 3.2

Substitute $- 9$ for $x$ in $y = \frac{2}{3} \cdot (- 9) + 5$ and solve for $y$ .

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Step 3.2.1

Multiply $\frac{2}{3}$ by $- 9$ .

$y = \frac{2}{3} \cdot - 9 + 5$

Step 3.2.2

Simplify $\frac{2}{3} \cdot - 9 + 5$ .

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1.1

Factor $3$ out of $- 9$ .

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

Step 3.2.2.1.1.2

Cancel the common factor.

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

Step 3.2.2.1.1.3

Rewrite the expression.

$y = 2 \cdot - 3 + 5$

Step 3.2.2.1.2

Multiply $2$ by $- 3$ .

$y = - 6 + 5$

Step 3.2.2.2

Add $- 6$ and $5$ .

$y = - 1$

Step 4

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

$(- 9, - 1)$

Step 5

The result can be shown in multiple forms.

Point Form:

$(- 9, - 1)$

Equation Form:

$x = - 9, y = - 1$

Step 6