Algebra Examples

$(\frac{2}{3}, \frac{4}{6})$ , $(- \frac{1}{3}, \frac{5}{6})$

Step 1

Find the slope of the line between $(\frac{2}{3}, \frac{4}{6})$ and $(- \frac{1}{3}, \frac{5}{6})$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

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Step 1.1

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

Step 1.2

The change in $x$ is equal to the difference in x-coordinates (also called run), and the change in $y$ is equal to the difference in y-coordinates (also called rise).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Step 1.3

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{\frac{5}{6} - (\frac{4}{6})}{- \frac{1}{3} - (\frac{2}{3})}$

Step 1.4

Simplify.

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Step 1.4.1

Cancel the common factor of $4$ and $6$ .

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Step 1.4.1.1

Factor $2$ out of $4$ .

$m = \frac{\frac{5}{6} - \frac{2 (2)}{6}}{- \frac{1}{3} - (\frac{2}{3})}$

Step 1.4.1.2

Cancel the common factors.

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Step 1.4.1.2.1

Factor $2$ out of $6$ .

$m = \frac{\frac{5}{6} - \frac{2 \cdot 2}{2 \cdot 3}}{- \frac{1}{3} - (\frac{2}{3})}$

Step 1.4.1.2.2

Cancel the common factor.

$m = \frac{\frac{5}{6} - \frac{2 \cdot 2}{2 \cdot 3}}{- \frac{1}{3} - (\frac{2}{3})}$

Step 1.4.1.2.3

Rewrite the expression.

$m = \frac{\frac{5}{6} - \frac{2}{3}}{- \frac{1}{3} - (\frac{2}{3})}$

Step 1.4.2

Multiply the numerator and denominator of the fraction by $6$ .

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Step 1.4.2.1

Multiply $\frac{\frac{5}{6} - \frac{2}{3}}{- \frac{1}{3} - \frac{2}{3}}$ by $\frac{6}{6}$ .

$m = \frac{6}{6} \cdot \frac{\frac{5}{6} - \frac{2}{3}}{- \frac{1}{3} - \frac{2}{3}}$

Step 1.4.2.2

Combine.

$m = \frac{6 (\frac{5}{6} - \frac{2}{3})}{6 (- \frac{1}{3} - \frac{2}{3})}$

Step 1.4.3

Apply the distributive property.

$m = \frac{6 (\frac{5}{6}) + 6 (- \frac{2}{3})}{6 (- \frac{1}{3}) + 6 (- \frac{2}{3})}$

Step 1.4.4

Simplify by cancelling.

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Step 1.4.4.1

Cancel the common factor of $6$ .

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Step 1.4.4.1.1

Cancel the common factor.

$m = \frac{6 (\frac{5}{6}) + 6 (- \frac{2}{3})}{6 (- \frac{1}{3}) + 6 (- \frac{2}{3})}$

Step 1.4.4.1.2

Rewrite the expression.

$m = \frac{5 + 6 (- \frac{2}{3})}{6 (- \frac{1}{3}) + 6 (- \frac{2}{3})}$

Step 1.4.4.2

Cancel the common factor of $3$ .

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Step 1.4.4.2.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$m = \frac{5 + 6 (\frac{- 2}{3})}{6 (- \frac{1}{3}) + 6 (- \frac{2}{3})}$

Step 1.4.4.2.2

Factor $3$ out of $6$ .

$m = \frac{5 + 3 (2) (\frac{- 2}{3})}{6 (- \frac{1}{3}) + 6 (- \frac{2}{3})}$

Step 1.4.4.2.3

Cancel the common factor.

$m = \frac{5 + 3 \cdot (2 (\frac{- 2}{3}))}{6 (- \frac{1}{3}) + 6 (- \frac{2}{3})}$

Step 1.4.4.2.4

Rewrite the expression.

$m = \frac{5 + 2 \cdot - 2}{6 (- \frac{1}{3}) + 6 (- \frac{2}{3})}$

Step 1.4.4.3

Multiply $2$ by $- 2$ .

$m = \frac{5 - 4}{6 (- \frac{1}{3}) + 6 (- \frac{2}{3})}$

Step 1.4.4.4

Cancel the common factor of $3$ .

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Step 1.4.4.4.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$m = \frac{5 - 4}{6 (\frac{- 1}{3}) + 6 (- \frac{2}{3})}$

Step 1.4.4.4.2

Factor $3$ out of $6$ .

$m = \frac{5 - 4}{3 (2) (\frac{- 1}{3}) + 6 (- \frac{2}{3})}$

Step 1.4.4.4.3

Cancel the common factor.

$m = \frac{5 - 4}{3 \cdot (2 (\frac{- 1}{3})) + 6 (- \frac{2}{3})}$

Step 1.4.4.4.4

Rewrite the expression.

$m = \frac{5 - 4}{2 \cdot - 1 + 6 (- \frac{2}{3})}$

Step 1.4.4.5

Multiply $2$ by $- 1$ .

$m = \frac{5 - 4}{- 2 + 6 (- \frac{2}{3})}$

Step 1.4.4.6

Cancel the common factor of $3$ .

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Step 1.4.4.6.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$m = \frac{5 - 4}{- 2 + 6 (\frac{- 2}{3})}$

Step 1.4.4.6.2

Factor $3$ out of $6$ .

$m = \frac{5 - 4}{- 2 + 3 (2) (\frac{- 2}{3})}$

Step 1.4.4.6.3

Cancel the common factor.

$m = \frac{5 - 4}{- 2 + 3 \cdot (2 (\frac{- 2}{3}))}$

Step 1.4.4.6.4

Rewrite the expression.

$m = \frac{5 - 4}{- 2 + 2 \cdot - 2}$

Step 1.4.4.7

Multiply $2$ by $- 2$ .

$m = \frac{5 - 4}{- 2 - 4}$

Step 1.4.5

Simplify the expression.

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Step 1.4.5.1

Subtract $4$ from $5$ .

$m = \frac{1}{- 2 - 4}$

Step 1.4.5.2

Subtract $4$ from $- 2$ .

$m = \frac{1}{- 6}$

Step 1.4.5.3

Move the negative in front of the fraction.

$m = - \frac{1}{6}$

Step 2

Use the slope $- \frac{1}{6}$ and a given point $(\frac{2}{3}, \frac{4}{6})$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{4}{6}) = - \frac{1}{6} \cdot (x - (\frac{2}{3}))$

Step 3

Simplify the equation and keep it in point-slope form.

$y - \frac{2}{3} = - \frac{1}{6} \cdot (x - \frac{2}{3})$

Step 4

Solve for $y$ .

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Step 4.1

Simplify $- \frac{1}{6} \cdot (x - \frac{2}{3})$ .

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Step 4.1.1

Rewrite.

$y - \frac{2}{3} = 0 + 0 - \frac{1}{6} \cdot (x - \frac{2}{3})$

Step 4.1.2

Simplify by adding zeros.

$y - \frac{2}{3} = - \frac{1}{6} \cdot (x - \frac{2}{3})$

Step 4.1.3

Apply the distributive property.

$y - \frac{2}{3} = - \frac{1}{6} x - \frac{1}{6} (- \frac{2}{3})$

Step 4.1.4

Combine $x$ and $\frac{1}{6}$ .

$y - \frac{2}{3} = - \frac{x}{6} - \frac{1}{6} (- \frac{2}{3})$

Step 4.1.5

Cancel the common factor of $2$ .

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Step 4.1.5.1

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$y - \frac{2}{3} = - \frac{x}{6} + \frac{- 1}{6} (- \frac{2}{3})$

Step 4.1.5.2

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$y - \frac{2}{3} = - \frac{x}{6} + \frac{- 1}{6} \cdot \frac{- 2}{3}$

Step 4.1.5.3

Factor $2$ out of $6$ .

$y - \frac{2}{3} = - \frac{x}{6} + \frac{- 1}{2 (3)} \cdot \frac{- 2}{3}$

Step 4.1.5.4

Factor $2$ out of $- 2$ .

$y - \frac{2}{3} = - \frac{x}{6} + \frac{- 1}{2 \cdot 3} \cdot \frac{2 \cdot - 1}{3}$

Step 4.1.5.5

Cancel the common factor.

$y - \frac{2}{3} = - \frac{x}{6} + \frac{- 1}{2 \cdot 3} \cdot \frac{2 \cdot - 1}{3}$

Step 4.1.5.6

Rewrite the expression.

$y - \frac{2}{3} = - \frac{x}{6} + \frac{- 1}{3} \cdot \frac{- 1}{3}$

Step 4.1.6

Multiply $\frac{- 1}{3}$ by $\frac{- 1}{3}$ .

$y - \frac{2}{3} = - \frac{x}{6} + \frac{- - 1}{3 \cdot 3}$

Step 4.1.7

Multiply.

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Step 4.1.7.1

Multiply $- 1$ by $- 1$ .

$y - \frac{2}{3} = - \frac{x}{6} + \frac{1}{3 \cdot 3}$

Step 4.1.7.2

Multiply $3$ by $3$ .

$y - \frac{2}{3} = - \frac{x}{6} + \frac{1}{9}$

Step 4.2

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

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Step 4.2.1

Add $\frac{2}{3}$ to both sides of the equation.

$y = - \frac{x}{6} + \frac{1}{9} + \frac{2}{3}$

Step 4.2.2

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = - \frac{x}{6} + \frac{1}{9} + \frac{2}{3} \cdot \frac{3}{3}$

Step 4.2.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.3.1

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

$y = - \frac{x}{6} + \frac{1}{9} + \frac{2 \cdot 3}{3 \cdot 3}$

Step 4.2.3.2

Multiply $3$ by $3$ .

$y = - \frac{x}{6} + \frac{1}{9} + \frac{2 \cdot 3}{9}$

Step 4.2.4

Combine the numerators over the common denominator.

$y = - \frac{x}{6} + \frac{1 + 2 \cdot 3}{9}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $2$ by $3$ .

$y = - \frac{x}{6} + \frac{1 + 6}{9}$

Step 4.2.5.2

Add $1$ and $6$ .

$y = - \frac{x}{6} + \frac{7}{9}$

Step 5

Reorder terms.

$y = - (\frac{1}{6} x) + \frac{7}{9}$

Step 6

Remove parentheses.

$y = - \frac{1}{6} x + \frac{7}{9}$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{1}{6} x + \frac{7}{9}$

Point-slope form:

$y - \frac{2}{3} = - \frac{1}{6} \cdot (x - \frac{2}{3})$

Step 8