Algebra Examples

$y = \frac{5 x + 1}{3}$ ;

$(1, 1)$

Step 1

Find the slope when

$y = \frac{5 x + 1}{3}$ .

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Step 1.1

Rewrite in slope-intercept form.

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Step 1.1.1

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 1.1.2

Split the fraction

$\frac{5 x + 1}{3}$ into two fractions.

$y = \frac{5 x}{3} + \frac{1}{3}$

Step 1.1.3

Reorder terms.

$y = \frac{5}{3} x + \frac{1}{3}$

Step 1.2

Using the slope-intercept form, the slope is

$\frac{5}{3}$ .

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

Step 2

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

$m_{perpendicular} = - \frac{1}{\frac{5}{3}}$

Step 3

Simplify

$- \frac{1}{\frac{5}{3}}$ to find the slope of the perpendicular line.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{3}{5}))$

Step 3.2

Multiply

$\frac{3}{5}$ by

$1$ .

$m_{perpendicular} = - \frac{3}{5}$

Step 4

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

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Step 4.1

Use the slope

$- \frac{3}{5}$ and a given point

$(1, 1)$ 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

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = - \frac{3}{5} \cdot (x - (1))$

Step 4.2

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

$y - 1 = - \frac{3}{5} \cdot (x - 1)$

Step 5

Write in

$y = m x + b$ form.

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Step 5.1

Solve for

$y$ .

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Step 5.1.1

Simplify

$- \frac{3}{5} \cdot (x - 1)$ .

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Step 5.1.1.1

Rewrite.

$y - 1 = 0 + 0 - \frac{3}{5} \cdot (x - 1)$

Step 5.1.1.2

Simplify by adding zeros.

$y - 1 = - \frac{3}{5} \cdot (x - 1)$

Step 5.1.1.3

Apply the distributive property.

$y - 1 = - \frac{3}{5} x - \frac{3}{5} \cdot - 1$

Step 5.1.1.4

Combine

$x$ and

$\frac{3}{5}$ .

$y - 1 = - \frac{x \cdot 3}{5} - \frac{3}{5} \cdot - 1$

Step 5.1.1.5

Multiply

$- \frac{3}{5} \cdot - 1$ .

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Step 5.1.1.5.1

Multiply

$- 1$ by

$- 1$ .

$y - 1 = - \frac{x \cdot 3}{5} + 1 (\frac{3}{5})$

Step 5.1.1.5.2

Multiply

$\frac{3}{5}$ by

$1$ .

$y - 1 = - \frac{x \cdot 3}{5} + \frac{3}{5}$

Step 5.1.1.6

Move

$3$ to the left of

$x$ .

$y - 1 = - \frac{3 x}{5} + \frac{3}{5}$

Step 5.1.2

Move all terms not containing

$y$ to the right side of the equation.

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Step 5.1.2.1

Add

$1$ to both sides of the equation.

$y = - \frac{3 x}{5} + \frac{3}{5} + 1$

Step 5.1.2.2

Write

$1$ as a fraction with a common denominator.

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

Step 5.1.2.3

Combine the numerators over the common denominator.

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

Step 5.1.2.4

Add

$3$ and

$5$ .

$y = - \frac{3 x}{5} + \frac{8}{5}$

Step 5.2

Reorder terms.

$y = - (\frac{3}{5} x) + \frac{8}{5}$

Step 5.3

Remove parentheses.

$y = - \frac{3}{5} x + \frac{8}{5}$

Step 6