Algebra Examples

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

Step 1

Find the slope of the line between $(- \frac{1}{2}, \frac{1}{2})$ and $(\frac{1}{4}, \frac{3}{4})$ 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.

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

Step 1.4

Simplify.

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Step 1.4.1

Multiply the numerator and denominator of the fraction by $4$ .

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Step 1.4.1.1

Multiply $\frac{\frac{3}{4} - \frac{1}{2}}{\frac{1}{4} - - \frac{1}{2}}$ by $\frac{4}{4}$ .

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

Step 1.4.1.2

Combine.

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

Step 1.4.2

Apply the distributive property.

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

Step 1.4.3

Simplify by cancelling.

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Step 1.4.3.1

Cancel the common factor of $4$ .

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Step 1.4.3.1.1

Cancel the common factor.

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

Step 1.4.3.1.2

Rewrite the expression.

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

Step 1.4.3.2

Cancel the common factor of $2$ .

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Step 1.4.3.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 1.4.3.2.2

Factor $2$ out of $4$ .

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

Step 1.4.3.2.3

Cancel the common factor.

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

Step 1.4.3.2.4

Rewrite the expression.

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

Step 1.4.3.3

Multiply $2$ by $- 1$ .

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

Step 1.4.3.4

Cancel the common factor of $4$ .

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Step 1.4.3.4.1

Cancel the common factor.

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

Step 1.4.3.4.2

Rewrite the expression.

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

Step 1.4.4

Subtract $2$ from $3$ .

$m = \frac{1}{1 + 4 (\frac{1}{2})}$

Step 1.4.5

Simplify the denominator.

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Step 1.4.5.1

Multiply $- - \frac{1}{2}$ .

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Step 1.4.5.1.1

Multiply $- 1$ by $- 1$ .

$m = \frac{1}{1 + 4 (1 (\frac{1}{2}))}$

Step 1.4.5.1.2

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

$m = \frac{1}{1 + 4 (\frac{1}{2})}$

Step 1.4.5.2

Cancel the common factor of $2$ .

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Step 1.4.5.2.1

Factor $2$ out of $4$ .

$m = \frac{1}{1 + 2 (2) (\frac{1}{2})}$

Step 1.4.5.2.2

Cancel the common factor.

$m = \frac{1}{1 + 2 \cdot (2 (\frac{1}{2}))}$

Step 1.4.5.2.3

Rewrite the expression.

$m = \frac{1}{1 + 2}$

Step 1.4.5.3

Add $1$ and $2$ .

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

Step 2

Use the slope $\frac{1}{3}$ and a given point $(- \frac{1}{2}, \frac{1}{2})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 3

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

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

Step 4

Solve for $y$ .

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Step 4.1

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

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Step 4.1.1

Rewrite.

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

Step 4.1.2

Simplify by adding zeros.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.1.4

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

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

Step 4.1.5

Multiply $\frac{1}{3} \cdot \frac{1}{2}$ .

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Step 4.1.5.1

Multiply $\frac{1}{3}$ by $\frac{1}{2}$ .

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

Step 4.1.5.2

Multiply $3$ by $2$ .

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

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 $\frac{1}{2}$ to both sides of the equation.

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

Step 4.2.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4.2.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.3.1

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

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

Step 4.2.3.2

Multiply $2$ by $3$ .

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

Step 4.2.4

Combine the numerators over the common denominator.

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

Step 4.2.5

Add $1$ and $3$ .

$y = \frac{x}{3} + \frac{4}{6}$

Step 4.2.6

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

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Step 4.2.6.1

Factor $2$ out of $4$ .

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

Step 4.2.6.2

Cancel the common factors.

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Step 4.2.6.2.1

Factor $2$ out of $6$ .

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

Step 4.2.6.2.2

Cancel the common factor.

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

Step 4.2.6.2.3

Rewrite the expression.

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

Step 5

Reorder terms.

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

Step 6

List the equation in different forms.

Slope-intercept form:

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

Point-slope form:

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

Step 7