Find the Equation Using Point-Slope Formula (5,4) , ( square root of 3,7)

$(5, 4)$ , $(\sqrt{3}, 7)$

Step 1

Find the slope of the line between $(5, 4)$ and $(\sqrt{3}, 7)$ 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.

$m = \frac{7 - (4)}{\sqrt{3} - (5)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $4$ .

$m = \frac{7 - 4}{\sqrt{3} - (5)}$

Subtract $4$ from $7$ .

$m = \frac{3}{\sqrt{3} - (5)}$

Multiply $- 1$ by $5$ .

$m = \frac{3}{\sqrt{3} - 5}$

Multiply $\frac{3}{\sqrt{3} - 5}$ by $\frac{\sqrt{3} + 5}{\sqrt{3} + 5}$ .

$m = \frac{3}{\sqrt{3} - 5} \cdot \frac{\sqrt{3} + 5}{\sqrt{3} + 5}$

Multiply $\frac{3}{\sqrt{3} - 5}$ by $\frac{\sqrt{3} + 5}{\sqrt{3} + 5}$ .

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

Expand the denominator using the FOIL method.

$m = \frac{3 (\sqrt{3} + 5)}{{\sqrt{3}}^{2} + \sqrt{3} \cdot 5 - 5 \sqrt{3} - 25}$

Simplify.

$m = \frac{3 (\sqrt{3} + 5)}{- 22}$

Move the negative in front of the fraction.

$m = - \frac{3 (\sqrt{3} + 5)}{22}$

Step 2

Use the slope $- \frac{3 (\sqrt{3} + 5)}{22}$ and a given point $(5, 4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (4) = - \frac{3 (\sqrt{3} + 5)}{22} \cdot (x - (5))$

Step 3

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

$y - 4 = - \frac{3 (\sqrt{3} + 5)}{22} \cdot (x - 5)$

Step 4

Solve for $y$ .

Simplify $- \frac{3 (\sqrt{3} + 5)}{22} \cdot (x - 5)$ .

Rewrite.

$y - 4 = 0 + 0 - \frac{3 (\sqrt{3} + 5)}{22} \cdot (x - 5)$

Simplify by adding zeros.

$y - 4 = - \frac{3 (\sqrt{3} + 5)}{22} \cdot (x - 5)$

Apply the distributive property.

$y - 4 = - \frac{3 (\sqrt{3} + 5)}{22} x - \frac{3 (\sqrt{3} + 5)}{22} \cdot - 5$

Combine $x$ and $\frac{3 (\sqrt{3} + 5)}{22}$ .

$y - 4 = - \frac{x (3 (\sqrt{3} + 5))}{22} - \frac{3 (\sqrt{3} + 5)}{22} \cdot - 5$

Multiply $- \frac{3 (\sqrt{3} + 5)}{22} \cdot - 5$ .

Multiply $- 5$ by $- 1$ .

$y - 4 = - \frac{x (3 (\sqrt{3} + 5))}{22} + 5 \frac{3 (\sqrt{3} + 5)}{22}$

Combine $5$ and $\frac{3 (\sqrt{3} + 5)}{22}$ .

$y - 4 = - \frac{x (3 (\sqrt{3} + 5))}{22} + \frac{5 (3 (\sqrt{3} + 5))}{22}$

Multiply $3$ by $5$ .

$y - 4 = - \frac{x (3 (\sqrt{3} + 5))}{22} + \frac{15 (\sqrt{3} + 5)}{22}$

Simplify terms.

Move $3$ to the left of $x$ .

$y - 4 = - \frac{3 x (\sqrt{3} + 5)}{22} + \frac{15 (\sqrt{3} + 5)}{22}$

Combine the numerators over the common denominator.

$y - 4 = \frac{- 3 x (\sqrt{3} + 5) + 15 (\sqrt{3} + 5)}{22}$

Simplify the numerator.

Apply the distributive property.

$y - 4 = \frac{- 3 x \sqrt{3} - 3 x \cdot 5 + 15 (\sqrt{3} + 5)}{22}$

Multiply $5$ by $- 3$ .

$y - 4 = \frac{- 3 x \sqrt{3} - 15 x + 15 (\sqrt{3} + 5)}{22}$

Apply the distributive property.

$y - 4 = \frac{- 3 x \sqrt{3} - 15 x + 15 \sqrt{3} + 15 \cdot 5}{22}$

Multiply $15$ by $5$ .

$y - 4 = \frac{- 3 x \sqrt{3} - 15 x + 15 \sqrt{3} + 75}{22}$

Reorder terms.

$y - 4 = \frac{- 3 x \sqrt{3} - 15 x + 75 + 15 \sqrt{3}}{22}$

Factor $3$ out of $- 3 x \sqrt{3} - 15 x + 75 + 15 \sqrt{3}$ .

Factor $3$ out of $- 3 x \sqrt{3}$ .

$y - 4 = \frac{3 (- x \sqrt{3}) - 15 x + 75 + 15 \sqrt{3}}{22}$

Factor $3$ out of $- 15 x$ .

$y - 4 = \frac{3 (- x \sqrt{3}) + 3 (- 5 x) + 75 + 15 \sqrt{3}}{22}$

Factor $3$ out of $75$ .

$y - 4 = \frac{3 (- x \sqrt{3}) + 3 (- 5 x) + 3 (25) + 15 \sqrt{3}}{22}$

Factor $3$ out of $15 \sqrt{3}$ .

$y - 4 = \frac{3 (- x \sqrt{3}) + 3 (- 5 x) + 3 (25) + 3 (5 \sqrt{3})}{22}$

Factor $3$ out of $3 (- x \sqrt{3}) + 3 (- 5 x)$ .

$y - 4 = \frac{3 (- x \sqrt{3} - 5 x) + 3 (25) + 3 (5 \sqrt{3})}{22}$

Factor $3$ out of $3 (- x \sqrt{3} - 5 x) + 3 (25)$ .

$y - 4 = \frac{3 (- x \sqrt{3} - 5 x + 25) + 3 (5 \sqrt{3})}{22}$

Factor $3$ out of $3 (- x \sqrt{3} - 5 x + 25) + 3 (5 \sqrt{3})$ .

$y - 4 = \frac{3 (- x \sqrt{3} - 5 x + 25 + 5 \sqrt{3})}{22}$

Simplify with factoring out.

Factor $- 1$ out of $- x \sqrt{3}$ .

$y - 4 = \frac{3 (- (x \sqrt{3}) - 5 x + 25 + 5 \sqrt{3})}{22}$

Factor $- 1$ out of $- 5 x$ .

$y - 4 = \frac{3 (- (x \sqrt{3}) - (5 x) + 25 + 5 \sqrt{3})}{22}$

Factor $- 1$ out of $- (x \sqrt{3}) - (5 x)$ .

$y - 4 = \frac{3 (- (x \sqrt{3} + 5 x) + 25 + 5 \sqrt{3})}{22}$

Rewrite $25$ as $- 1 (- 25)$ .

$y - 4 = \frac{3 (- (x \sqrt{3} + 5 x) - 1 (- 25) + 5 \sqrt{3})}{22}$

Factor $- 1$ out of $- (x \sqrt{3} + 5 x) - 1 (- 25)$ .

$y - 4 = \frac{3 (- (x \sqrt{3} + 5 x - 25) + 5 \sqrt{3})}{22}$

Factor $- 1$ out of $5 \sqrt{3}$ .

$y - 4 = \frac{3 (- (x \sqrt{3} + 5 x - 25) - (- 5 \sqrt{3}))}{22}$

Factor $- 1$ out of $- (x \sqrt{3} + 5 x - 25) - (- 5 \sqrt{3})$ .

$y - 4 = \frac{3 (- (x \sqrt{3} + 5 x - 25 - 5 \sqrt{3}))}{22}$

Simplify the expression.

Rewrite $- (x \sqrt{3} + 5 x - 25 - 5 \sqrt{3})$ as $- 1 (x \sqrt{3} + 5 x - 25 - 5 \sqrt{3})$ .

$y - 4 = \frac{3 (- 1 (x \sqrt{3} + 5 x - 25 - 5 \sqrt{3}))}{22}$

Move the negative in front of the fraction.

$y - 4 = - \frac{3 (x \sqrt{3} + 5 x - 25 - 5 \sqrt{3})}{22}$

Add $4$ to both sides of the equation.

$y = - \frac{3 (x \sqrt{3} + 5 x - 25 - 5 \sqrt{3})}{22} + 4$

Step 5

Reorder terms.

$y = - (\frac{3}{22} (x \sqrt{3} + 5 x - 25 - 5 \sqrt{3})) + 4$

Step 6

Remove parentheses.

$y = - \frac{3}{22} (x \sqrt{3} + 5 x - 25 - 5 \sqrt{3}) + 4$

Step 7

List the equation in different forms.

Slope-intercept form:

$y = - \frac{3}{22} \cdot (x \sqrt{3} + 5 x - 25 - 5 \sqrt{3}) + 4$

Point-slope form:

$y - 4 = - \frac{3 (\sqrt{3} + 5)}{22} \cdot (x - 5)$

Step 8

Find the Equation Using Point-Slope Formula (5,4) , ( square root of 3,7)

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