Algebra Examples

$y - 4 = \frac{5}{2} (x + 3)$ ; $(- 7, 8)$

Step 1

Solve $y - 4 = \frac{5}{2} \cdot (x + 3)$ .

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Step 1.1

Simplify $\frac{5}{2} (x + 3)$ .

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Step 1.1.1

Rewrite.

$y - 4 = 0 + 0 + \frac{5}{2} (x + 3)$

Step 1.1.2

Simplify by adding zeros.

$y - 4 = \frac{5}{2} (x + 3)$

Step 1.1.3

Apply the distributive property.

$y - 4 = \frac{5}{2} x + \frac{5}{2} \cdot 3$

Step 1.1.4

Combine $\frac{5}{2}$ and $x$ .

$y - 4 = \frac{5 x}{2} + \frac{5}{2} \cdot 3$

Step 1.1.5

Multiply $\frac{5}{2} \cdot 3$ .

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Step 1.1.5.1

Combine $\frac{5}{2}$ and $3$ .

$y - 4 = \frac{5 x}{2} + \frac{5 \cdot 3}{2}$

Step 1.1.5.2

Multiply $5$ by $3$ .

$y - 4 = \frac{5 x}{2} + \frac{15}{2}$

Step 1.2

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

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Step 1.2.1

Add $4$ to both sides of the equation.

$y = \frac{5 x}{2} + \frac{15}{2} + 4$

Step 1.2.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{5 x}{2} + \frac{15}{2} + 4 \cdot \frac{2}{2}$

Step 1.2.3

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

$y = \frac{5 x}{2} + \frac{15}{2} + \frac{4 \cdot 2}{2}$

Step 1.2.4

Combine the numerators over the common denominator.

$y = \frac{5 x}{2} + \frac{15 + 4 \cdot 2}{2}$

Step 1.2.5

Simplify the numerator.

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Step 1.2.5.1

Multiply $4$ by $2$ .

$y = \frac{5 x}{2} + \frac{15 + 8}{2}$

Step 1.2.5.2

Add $15$ and $8$ .

$y = \frac{5 x}{2} + \frac{23}{2}$

Step 2

Find the slope when $y = \frac{5 x}{2} + \frac{23}{2}$ .

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Step 2.1

Rewrite in slope-intercept form.

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Step 2.1.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 2.1.2

Reorder terms.

$y = \frac{5}{2} x + \frac{23}{2}$

Step 2.2

Using the slope-intercept form, the slope is $\frac{5}{2}$ .

$m = \frac{5}{2}$

Step 3

The equation of a perpendicular line must have a slope that is the negative reciprocal of the original slope.

$m_{perpendicular} = - \frac{1}{\frac{5}{2}}$

Step 4

Simplify $- \frac{1}{\frac{5}{2}}$ to find the slope of the perpendicular line.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{2}{5}))$

Step 4.2

Multiply $\frac{2}{5}$ by $1$ .

$m_{perpendicular} = - \frac{2}{5}$

Step 5

Find the equation of the perpendicular line using the point-slope formula.

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Step 5.1

Use the slope $- \frac{2}{5}$ and a given point $(- 7, 8)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (8) = - \frac{2}{5} \cdot (x - (- 7))$

Step 5.2

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

$y - 8 = - \frac{2}{5} \cdot (x + 7)$

Step 6

Write in $y = m x + b$ form.

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Step 6.1

Solve for $y$ .

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Step 6.1.1

Simplify $- \frac{2}{5} \cdot (x + 7)$ .

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Step 6.1.1.1

Rewrite.

$y - 8 = 0 + 0 - \frac{2}{5} \cdot (x + 7)$

Step 6.1.1.2

Simplify by adding zeros.

$y - 8 = - \frac{2}{5} \cdot (x + 7)$

Step 6.1.1.3

Apply the distributive property.

$y - 8 = - \frac{2}{5} x - \frac{2}{5} \cdot 7$

Step 6.1.1.4

Combine $x$ and $\frac{2}{5}$ .

$y - 8 = - \frac{x \cdot 2}{5} - \frac{2}{5} \cdot 7$

Step 6.1.1.5

Multiply $- \frac{2}{5} \cdot 7$ .

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Step 6.1.1.5.1

Multiply $7$ by $- 1$ .

$y - 8 = - \frac{x \cdot 2}{5} - 7 (\frac{2}{5})$

Step 6.1.1.5.2

Combine $- 7$ and $\frac{2}{5}$ .

$y - 8 = - \frac{x \cdot 2}{5} + \frac{- 7 \cdot 2}{5}$

Step 6.1.1.5.3

Multiply $- 7$ by $2$ .

$y - 8 = - \frac{x \cdot 2}{5} + \frac{- 14}{5}$

Step 6.1.1.6

Simplify each term.

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Step 6.1.1.6.1

Move $2$ to the left of $x$ .

$y - 8 = - \frac{2 \cdot x}{5} + \frac{- 14}{5}$

Step 6.1.1.6.2

Move the negative in front of the fraction.

$y - 8 = - \frac{2 x}{5} - \frac{14}{5}$

Step 6.1.2

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

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Step 6.1.2.1

Add $8$ to both sides of the equation.

$y = - \frac{2 x}{5} - \frac{14}{5} + 8$

Step 6.1.2.2

To write $8$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = - \frac{2 x}{5} - \frac{14}{5} + 8 \cdot \frac{5}{5}$

Step 6.1.2.3

Combine $8$ and $\frac{5}{5}$ .

$y = - \frac{2 x}{5} - \frac{14}{5} + \frac{8 \cdot 5}{5}$

Step 6.1.2.4

Combine the numerators over the common denominator.

$y = - \frac{2 x}{5} + \frac{- 14 + 8 \cdot 5}{5}$

Step 6.1.2.5

Simplify the numerator.

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Step 6.1.2.5.1

Multiply $8$ by $5$ .

$y = - \frac{2 x}{5} + \frac{- 14 + 40}{5}$

Step 6.1.2.5.2

Add $- 14$ and $40$ .

$y = - \frac{2 x}{5} + \frac{26}{5}$

Step 6.2

Reorder terms.

$y = - (\frac{2}{5} x) + \frac{26}{5}$

Step 6.3

Remove parentheses.

$y = - \frac{2}{5} x + \frac{26}{5}$

Step 7