Find the Equation Using Point-Slope Formula (0,0) , (-1/6,1/7)

$(0, 0)$ , $(- \frac{1}{6}, \frac{1}{7})$

Step 1

Find the slope of the line between $(0, 0)$ and $(- \frac{1}{6}, \frac{1}{7})$ 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.

$m = \frac{\frac{1}{7} - (0)}{- \frac{1}{6} - (0)}$

Simplify.

Multiply the numerator and denominator of the complex fraction by $42$ .

Multiply $\frac{\frac{1}{7} - (0)}{- \frac{1}{6} - (0)}$ by $\frac{42}{42}$ .

$m = \frac{42}{42} \cdot \frac{\frac{1}{7} - (0)}{- \frac{1}{6} - (0)}$

Combine.

$m = \frac{42 (\frac{1}{7} - (0))}{42 (- \frac{1}{6} - (0))}$

Apply the distributive property.

$m = \frac{42 (\frac{1}{7}) + 42 (- (0))}{42 (- \frac{1}{6}) + 42 (- (0))}$

Simplify by cancelling.

Cancel the common factor of $7$ .

Factor $7$ out of $42$ .

$m = \frac{7 (6) (\frac{1}{7}) + 42 (- (0))}{42 (- \frac{1}{6}) + 42 (- (0))}$

Cancel the common factor.

$m = \frac{7 \cdot (6 (\frac{1}{7})) + 42 (- (0))}{42 (- \frac{1}{6}) + 42 (- (0))}$

Rewrite the expression.

$m = \frac{6 + 42 (- (0))}{42 (- \frac{1}{6}) + 42 (- (0))}$

Cancel the common factor of $6$ .

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

$m = \frac{6 + 42 (- (0))}{42 (\frac{- 1}{6}) + 42 (- (0))}$

Factor $6$ out of $42$ .

$m = \frac{6 + 42 (- (0))}{6 (7) (\frac{- 1}{6}) + 42 (- (0))}$

Cancel the common factor.

$m = \frac{6 + 42 (- (0))}{6 \cdot (7 (\frac{- 1}{6})) + 42 (- (0))}$

Rewrite the expression.

$m = \frac{6 + 42 (- (0))}{7 \cdot - 1 + 42 (- (0))}$

Multiply $7$ by $- 1$ .

$m = \frac{6 + 42 (- (0))}{- 7 + 42 (- (0))}$

Simplify the numerator.

Multiply $42 (- (0))$ .

Multiply $- 1$ by $0$ .

$m = \frac{6 + 42 \cdot 0}{- 7 + 42 (- (0))}$

Multiply $42$ by $0$ .

$m = \frac{6 + 0}{- 7 + 42 (- (0))}$

Add $6$ and $0$ .

$m = \frac{6}{- 7 + 42 (- (0))}$

Simplify the denominator.

Multiply $42 (- (0))$ .

Multiply $- 1$ by $0$ .

$m = \frac{6}{- 7 + 42 \cdot 0}$

Multiply $42$ by $0$ .

$m = \frac{6}{- 7 + 0}$

Add $- 7$ and $0$ .

$m = \frac{6}{- 7}$

Move the negative in front of the fraction.

$m = - \frac{6}{7}$

Step 2

Use the slope $- \frac{6}{7}$ and a given point $(0, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = - \frac{6}{7} \cdot (x - (0))$

Step 3

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

$y + 0 = - \frac{6}{7} \cdot (x + 0)$

Step 4

Solve for $y$ .

Add $y$ and $0$ .

$y = - \frac{6}{7} \cdot (x + 0)$

Simplify $- \frac{6}{7} \cdot (x + 0)$ .

Add $x$ and $0$ .

$y = - \frac{6}{7} \cdot x$

Combine $x$ and $\frac{6}{7}$ .

$y = - \frac{x \cdot 6}{7}$

Move $6$ to the left of $x$ .

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

Step 5

Reorder terms.

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

Step 6

Remove parentheses.

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

Step 7

List the equation in different forms.

Slope-intercept form:

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

Point-slope form:

$y + 0 = - \frac{6}{7} \cdot (x + 0)$

Step 8

Find the Equation Using Point-Slope Formula (0,0) , (-1/6,1/7)

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