Find the Equation Using Point-Slope Formula (-1/2,2) , (-4,2/3)

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

Step 1

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

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

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

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).

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

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

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

Simplify.

Multiply the numerator and denominator of the complex fraction by $3$ .

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

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

Combine.

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

Apply the distributive property.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the numerator.

Multiply $3 (- (2))$ .

Multiply $- 1$ by $2$ .

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

Multiply $3$ by $- 2$ .

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

Subtract $6$ from $2$ .

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

Simplify the denominator.

Multiply $3$ by $- 4$ .

$m = \frac{- 4}{- 12 + 3 (\frac{1}{2})}$

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

Multiply $- 1$ by $- 1$ .

$m = \frac{- 4}{- 12 + 3 (1 (\frac{1}{2}))}$

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

$m = \frac{- 4}{- 12 + 3 (\frac{1}{2})}$

Combine $3$ and $\frac{1}{2}$ .

$m = \frac{- 4}{- 12 + \frac{3}{2}}$

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$m = \frac{- 4}{- 12 \cdot \frac{2}{2} + \frac{3}{2}}$

Combine $- 12$ and $\frac{2}{2}$ .

$m = \frac{- 4}{\frac{- 12 \cdot 2}{2} + \frac{3}{2}}$

Combine the numerators over the common denominator.

$m = \frac{- 4}{\frac{- 12 \cdot 2 + 3}{2}}$

Simplify the numerator.

Multiply $- 12$ by $2$ .

$m = \frac{- 4}{\frac{- 24 + 3}{2}}$

Add $- 24$ and $3$ .

$m = \frac{- 4}{\frac{- 21}{2}}$

Move the negative in front of the fraction.

$m = \frac{- 4}{- \frac{21}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$m = - 4 (- \frac{2}{21})$

Multiply $- 4 (- \frac{2}{21})$ .

Multiply $- 1$ by $- 4$ .

$m = 4 (\frac{2}{21})$

Combine $4$ and $\frac{2}{21}$ .

$m = \frac{4 \cdot 2}{21}$

Multiply $4$ by $2$ .

$m = \frac{8}{21}$

Step 2

Use the slope $\frac{8}{21}$ and a given point $(- \frac{1}{2}, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = \frac{8}{21} \cdot (x - (- \frac{1}{2}))$

Step 3

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

$y - 2 = \frac{8}{21} \cdot (x + \frac{1}{2})$

Step 4

Solve for $y$ .

Simplify $\frac{8}{21} \cdot (x + \frac{1}{2})$ .

Rewrite.

$y - 2 = 0 + 0 + \frac{8}{21} \cdot (x + \frac{1}{2})$

Simplify by adding zeros.

$y - 2 = \frac{8}{21} \cdot (x + \frac{1}{2})$

Apply the distributive property.

$y - 2 = \frac{8}{21} x + \frac{8}{21} \cdot \frac{1}{2}$

Combine $\frac{8}{21}$ and $x$ .

$y - 2 = \frac{8 x}{21} + \frac{8}{21} \cdot \frac{1}{2}$

Cancel the common factor of $2$ .

Factor $2$ out of $8$ .

$y - 2 = \frac{8 x}{21} + \frac{2 (4)}{21} \cdot \frac{1}{2}$

Cancel the common factor.

$y - 2 = \frac{8 x}{21} + \frac{2 \cdot 4}{21} \cdot \frac{1}{2}$

Rewrite the expression.

$y - 2 = \frac{8 x}{21} + \frac{4}{21}$

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

Add $2$ to both sides of the equation.

$y = \frac{8 x}{21} + \frac{4}{21} + 2$

To write $2$ as a fraction with a common denominator, multiply by $\frac{21}{21}$ .

$y = \frac{8 x}{21} + \frac{4}{21} + 2 \cdot \frac{21}{21}$

Combine $2$ and $\frac{21}{21}$ .

$y = \frac{8 x}{21} + \frac{4}{21} + \frac{2 \cdot 21}{21}$

Combine the numerators over the common denominator.

$y = \frac{8 x}{21} + \frac{4 + 2 \cdot 21}{21}$

Simplify the numerator.

Multiply $2$ by $21$ .

$y = \frac{8 x}{21} + \frac{4 + 42}{21}$

Add $4$ and $42$ .

$y = \frac{8 x}{21} + \frac{46}{21}$

Step 5

Reorder terms.

$y = \frac{8}{21} x + \frac{46}{21}$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{8}{21} x + \frac{46}{21}$

Point-slope form:

$y - 2 = \frac{8}{21} \cdot (x + \frac{1}{2})$

Step 7

Find the Equation Using Point-Slope Formula (-1/2,2) , (-4,2/3)

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