Algebra Examples

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

Step 1

Find the value of the slope.

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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.

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

Step 1.4

Simplify.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Multiply $- 1$ by $4$ .

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

Step 1.4.1.2

Subtract $4$ from $7$ .

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

Step 1.4.2

Multiply $- 1$ by $5$ .

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

Step 1.4.3

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

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

Step 1.4.4

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

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

Step 1.4.5

Expand the denominator using the FOIL method.

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

Step 1.4.6

Simplify.

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

Step 1.4.7

Move the negative in front of the fraction.

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

Step 2

Find the value of the y-intercept.

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Step 2.1

Substitute the value of $m$ into the slope-intercept form of the equation, $y = m x + b$ .

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

Step 2.2

Substitute the value of $x$ into the slope-intercept form of the equation, $y = m x + b$ .

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

Step 2.3

Substitute the value of $y$ into the slope-intercept form of the equation, $y = m x + b$ .

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

Step 2.4

Rewrite the equation as $(- \frac{3 (\sqrt{3} + 5)}{22}) \cdot (5) + b = 4$ .

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

Step 2.5

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

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

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

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Step 2.5.1.1.1

Multiply $5$ by $- 1$ .

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

Step 2.5.1.1.2

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

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

Step 2.5.1.1.3

Multiply $3$ by $- 5$ .

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

Step 2.5.1.2

Move the negative in front of the fraction.

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

Step 2.5.2

To write $b$ as a fraction with a common denominator, multiply by $\frac{22}{22}$ .

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

Step 2.5.3

Simplify terms.

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Step 2.5.3.1

Combine $b$ and $\frac{22}{22}$ .

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

Step 2.5.3.2

Combine the numerators over the common denominator.

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

Step 2.5.4

Simplify the numerator.

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Step 2.5.4.1

Apply the distributive property.

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

Step 2.5.4.2

Multiply $- 15$ by $5$ .

$\frac{- 15 \sqrt{3} - 75 + b \cdot 22}{22} = 4$

Step 2.5.4.3

Move $22$ to the left of $b$ .

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

Step 2.5.5

Simplify with factoring out.

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Step 2.5.5.1

Factor $- 1$ out of $- 15 \sqrt{3}$ .

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

Step 2.5.5.2

Rewrite $- 75$ as $- 1 (75)$ .

$\frac{- (15 \sqrt{3}) - 1 (75) + 22 b}{22} = 4$

Step 2.5.5.3

Factor $- 1$ out of $- (15 \sqrt{3}) - 1 (75)$ .

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

Step 2.5.5.4

Factor $- 1$ out of $22 b$ .

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

Step 2.5.5.5

Factor $- 1$ out of $- (15 \sqrt{3} + 75) - (- 22 b)$ .

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

Step 2.5.5.6

Simplify the expression.

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Step 2.5.5.6.1

Rewrite $- (15 \sqrt{3} + 75 - 22 b)$ as $- 1 (15 \sqrt{3} + 75 - 22 b)$ .

$\frac{- 1 (15 \sqrt{3} + 75 - 22 b)}{22} = 4$

Step 2.5.5.6.2

Move the negative in front of the fraction.

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

Step 2.6

Multiply both sides of the equation by $- 22$ .

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

Step 2.7

Simplify both sides of the equation.

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Step 2.7.1

Simplify the left side.

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Step 2.7.1.1

Simplify $- 22 (- \frac{15 \sqrt{3} + 75 - 22 b}{22})$ .

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Step 2.7.1.1.1

Cancel the common factor of $22$ .

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Step 2.7.1.1.1.1

Move the leading negative in $- \frac{15 \sqrt{3} + 75 - 22 b}{22}$ into the numerator.

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

Step 2.7.1.1.1.2

Factor $22$ out of $- 22$ .

$22 (- 1) \frac{- (15 \sqrt{3} + 75 - 22 b)}{22} = - 22 \cdot 4$

Step 2.7.1.1.1.3

Cancel the common factor.

$22 \cdot - 1 \frac{- (15 \sqrt{3} + 75 - 22 b)}{22} = - 22 \cdot 4$

Step 2.7.1.1.1.4

Rewrite the expression.

$- - (15 \sqrt{3} + 75 - 22 b) = - 22 \cdot 4$

Step 2.7.1.1.2

Multiply.

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Step 2.7.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (15 \sqrt{3} + 75 - 22 b) = - 22 \cdot 4$

Step 2.7.1.1.2.2

Multiply $15 \sqrt{3} + 75 - 22 b$ by $1$ .

$15 \sqrt{3} + 75 - 22 b = - 22 \cdot 4$

Step 2.7.2

Simplify the right side.

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Step 2.7.2.1

Multiply $- 22$ by $4$ .

$15 \sqrt{3} + 75 - 22 b = - 88$

Step 2.8

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

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Step 2.8.1

Subtract $15 \sqrt{3}$ from both sides of the equation.

$75 - 22 b = - 88 - 15 \sqrt{3}$

Step 2.8.2

Subtract $75$ from both sides of the equation.

$- 22 b = - 88 - 15 \sqrt{3} - 75$

Step 2.8.3

Subtract $75$ from $- 88$ .

$- 22 b = - 163 - 15 \sqrt{3}$

Step 2.9

Divide each term in $- 22 b = - 163 - 15 \sqrt{3}$ by $- 22$ and simplify.

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Step 2.9.1

Divide each term in $- 22 b = - 163 - 15 \sqrt{3}$ by $- 22$ .

$\frac{- 22 b}{- 22} = \frac{- 163}{- 22} + \frac{- 15 \sqrt{3}}{- 22}$

Step 2.9.2

Simplify the left side.

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Step 2.9.2.1

Cancel the common factor of $- 22$ .

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Step 2.9.2.1.1

Cancel the common factor.

$\frac{- 22 b}{- 22} = \frac{- 163}{- 22} + \frac{- 15 \sqrt{3}}{- 22}$

Step 2.9.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 163}{- 22} + \frac{- 15 \sqrt{3}}{- 22}$

Step 2.9.3

Simplify the right side.

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Step 2.9.3.1

Simplify each term.

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Step 2.9.3.1.1

Dividing two negative values results in a positive value.

$b = \frac{163}{22} + \frac{- 15 \sqrt{3}}{- 22}$

Step 2.9.3.1.2

Dividing two negative values results in a positive value.

$b = \frac{163}{22} + \frac{15 \sqrt{3}}{22}$

Step 3

List the slope and y-intercept.

Slope: $- \frac{3 (\sqrt{3} + 5)}{22}$

y-intercept: $(0, \frac{163}{22} + \frac{15 \sqrt{3}}{22})$

Step 4