Pre-Algebra Examples

$(- 2 \sqrt{2}, 0)$ , $(4, \sqrt{5})$

Step 1

Find the value of the slope.

Tap for more steps...

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.

$m = \frac{\sqrt{5} - (0)}{4 - (- 2 \sqrt{2})}$

Step 1.4

Simplify.

Tap for more steps...

Step 1.4.1

Simplify the numerator.

Tap for more steps...

Step 1.4.1.1

Multiply $- 1$ by $0$ .

$m = \frac{\sqrt{5} + 0}{4 - (- 2 \sqrt{2})}$

Step 1.4.1.2

Add $\sqrt{5}$ and $0$ .

$m = \frac{\sqrt{5}}{4 - (- 2 \sqrt{2})}$

Step 1.4.2

Multiply $- 2$ by $- 1$ .

$m = \frac{\sqrt{5}}{4 + 2 \sqrt{2}}$

Step 1.4.3

Multiply $\frac{\sqrt{5}}{4 + 2 \sqrt{2}}$ by $\frac{4 - 2 \sqrt{2}}{4 - 2 \sqrt{2}}$ .

$m = \frac{\sqrt{5}}{4 + 2 \sqrt{2}} \cdot \frac{4 - 2 \sqrt{2}}{4 - 2 \sqrt{2}}$

Step 1.4.4

Multiply $\frac{\sqrt{5}}{4 + 2 \sqrt{2}}$ by $\frac{4 - 2 \sqrt{2}}{4 - 2 \sqrt{2}}$ .

$m = \frac{\sqrt{5} (4 - 2 \sqrt{2})}{(4 + 2 \sqrt{2}) (4 - 2 \sqrt{2})}$

Step 1.4.5

Expand the denominator using the FOIL method.

$m = \frac{\sqrt{5} (4 - 2 \sqrt{2})}{16 - 8 \sqrt{2} + 8 \sqrt{2} - 4 {\sqrt{2}}^{2}}$

Step 1.4.6

Simplify.

$m = \frac{\sqrt{5} (4 - 2 \sqrt{2})}{8}$

Step 1.4.7

Cancel the common factor of $4 - 2 \sqrt{2}$ and $8$ .

Tap for more steps...

Step 1.4.7.1

Factor $2$ out of $\sqrt{5} (4 - 2 \sqrt{2})$ .

$m = \frac{2 (\sqrt{5} (2 - \sqrt{2}))}{8}$

Step 1.4.7.2

Cancel the common factors.

Tap for more steps...

Step 1.4.7.2.1

Factor $2$ out of $8$ .

$m = \frac{2 (\sqrt{5} (2 - \sqrt{2}))}{2 (4)}$

Step 1.4.7.2.2

Cancel the common factor.

$m = \frac{2 (\sqrt{5} (2 - \sqrt{2}))}{2 \cdot 4}$

Step 1.4.7.2.3

Rewrite the expression.

$m = \frac{\sqrt{5} (2 - \sqrt{2})}{4}$

Step 1.4.8

Apply the distributive property.

$m = \frac{\sqrt{5} \cdot 2 + \sqrt{5} (- \sqrt{2})}{4}$

Step 1.4.9

Move $2$ to the left of $\sqrt{5}$ .

$m = \frac{2 \cdot \sqrt{5} + \sqrt{5} (- \sqrt{2})}{4}$

Step 1.4.10

Multiply $\sqrt{5} (- \sqrt{2})$ .

Tap for more steps...

Step 1.4.10.1

Combine using the product rule for radicals.

$m = \frac{2 \cdot \sqrt{5} - \sqrt{5 \cdot 2}}{4}$

Step 1.4.10.2

Multiply $5$ by $2$ .

$m = \frac{2 \cdot \sqrt{5} - \sqrt{10}}{4}$

$m = \frac{2 \sqrt{5} - \sqrt{10}}{4}$

Step 2

Find the value of the y-intercept.

Tap for more steps...

Step 2.1

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

$y = (\frac{2 \sqrt{5} - \sqrt{10}}{4}) x + b$

Step 2.2

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

$y = (\frac{2 \sqrt{5} - \sqrt{10}}{4}) \cdot (- 2 \sqrt{2}) + b$

Step 2.3

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

$0 = (\frac{2 \sqrt{5} - \sqrt{10}}{4}) \cdot (- 2 \sqrt{2}) + b$

Step 2.4

Rewrite the equation as $(\frac{2 \sqrt{5} - \sqrt{10}}{4}) \cdot (- 2 \sqrt{2}) + b = 0$ .

$(\frac{2 \sqrt{5} - \sqrt{10}}{4}) \cdot (- 2 \sqrt{2}) + b = 0$

Step 2.5

Simplify each term.

Tap for more steps...

Step 2.5.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.5.1.1

Factor $2$ out of $4$ .

$\frac{2 \sqrt{5} - \sqrt{10}}{2 (2)} \cdot (- 2 \sqrt{2}) + b = 0$

Step 2.5.1.2

Factor $2$ out of $- 2 \sqrt{2}$ .

$\frac{2 \sqrt{5} - \sqrt{10}}{2 (2)} \cdot (2 (- \sqrt{2})) + b = 0$

Step 2.5.1.3

Cancel the common factor.

$\frac{2 \sqrt{5} - \sqrt{10}}{2 \cdot 2} \cdot (2 (- \sqrt{2})) + b = 0$

Step 2.5.1.4

Rewrite the expression.

$\frac{2 \sqrt{5} - \sqrt{10}}{2} \cdot (- \sqrt{2}) + b = 0$

Step 2.5.2

Combine $\frac{2 \sqrt{5} - \sqrt{10}}{2}$ and $\sqrt{2}$ .

$- \frac{(2 \sqrt{5} - \sqrt{10}) \sqrt{2}}{2} + b = 0$

Step 2.5.3

Apply the distributive property.

$- \frac{2 \sqrt{5} \sqrt{2} - \sqrt{10} \sqrt{2}}{2} + b = 0$

Step 2.5.4

Multiply $2 \sqrt{5} \sqrt{2}$ .

Tap for more steps...

Step 2.5.4.1

Combine using the product rule for radicals.

$- \frac{2 \sqrt{2 \cdot 5} - \sqrt{10} \sqrt{2}}{2} + b = 0$

Step 2.5.4.2

Multiply $2$ by $5$ .

$- \frac{2 \sqrt{10} - \sqrt{10} \sqrt{2}}{2} + b = 0$

Step 2.5.5

Multiply $- \sqrt{10} \sqrt{2}$ .

Tap for more steps...

Step 2.5.5.1

Combine using the product rule for radicals.

$- \frac{2 \sqrt{10} - \sqrt{2 \cdot 10}}{2} + b = 0$

Step 2.5.5.2

Multiply $2$ by $10$ .

$- \frac{2 \sqrt{10} - \sqrt{20}}{2} + b = 0$

Step 2.5.6

Simplify each term.

Tap for more steps...

Step 2.5.6.1

Rewrite $20$ as $2^{2} \cdot 5$ .

Tap for more steps...

Step 2.5.6.1.1

Factor $4$ out of $20$ .

$- \frac{2 \sqrt{10} - \sqrt{4 (5)}}{2} + b = 0$

Step 2.5.6.1.2

Rewrite $4$ as $2^{2}$ .

$- \frac{2 \sqrt{10} - \sqrt{2^{2} \cdot 5}}{2} + b = 0$

Step 2.5.6.2

Pull terms out from under the radical.

$- \frac{2 \sqrt{10} - (2 \sqrt{5})}{2} + b = 0$

Step 2.5.6.3

Multiply $2$ by $- 1$ .

$- \frac{2 \sqrt{10} - 2 \sqrt{5}}{2} + b = 0$

Step 2.5.7

Cancel the common factor of $2 \sqrt{10} - 2 \sqrt{5}$ and $2$ .

Tap for more steps...

Step 2.5.7.1

Factor $2$ out of $2 \sqrt{10}$ .

$- \frac{2 (\sqrt{10}) - 2 \sqrt{5}}{2} + b = 0$

Step 2.5.7.2

Factor $2$ out of $- 2 \sqrt{5}$ .

$- \frac{2 (\sqrt{10}) + 2 (- \sqrt{5})}{2} + b = 0$

Step 2.5.7.3

Factor $2$ out of $2 (\sqrt{10}) + 2 (- \sqrt{5})$ .

$- \frac{2 (\sqrt{10} - \sqrt{5})}{2} + b = 0$

Step 2.5.7.4

Cancel the common factors.

Tap for more steps...

Step 2.5.7.4.1

Factor $2$ out of $2$ .

$- \frac{2 (\sqrt{10} - \sqrt{5})}{2 (1)} + b = 0$

Step 2.5.7.4.2

Cancel the common factor.

$- \frac{2 (\sqrt{10} - \sqrt{5})}{2 \cdot 1} + b = 0$

Step 2.5.7.4.3

Rewrite the expression.

$- \frac{\sqrt{10} - \sqrt{5}}{1} + b = 0$

Step 2.5.7.4.4

Divide $\sqrt{10} - \sqrt{5}$ by $1$ .

$- (\sqrt{10} - \sqrt{5}) + b = 0$

Step 2.5.8

Apply the distributive property.

$- \sqrt{10} - - \sqrt{5} + b = 0$

Step 2.5.9

Multiply $- - \sqrt{5}$ .

Tap for more steps...

Step 2.5.9.1

Multiply $- 1$ by $- 1$ .

$- \sqrt{10} + 1 \sqrt{5} + b = 0$

Step 2.5.9.2

Multiply $\sqrt{5}$ by $1$ .

$- \sqrt{10} + \sqrt{5} + b = 0$

Step 2.6

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

Tap for more steps...

Step 2.6.1

Add $\sqrt{10}$ to both sides of the equation.

$\sqrt{5} + b = \sqrt{10}$

Step 2.6.2

Subtract $\sqrt{5}$ from both sides of the equation.

$b = \sqrt{10} - \sqrt{5}$

Step 3

List the slope and y-intercept.

Slope: $\frac{2 \sqrt{5} - \sqrt{10}}{4}$

y-intercept: $(0, \sqrt{10} - \sqrt{5})$

Step 4