Algebra Examples

$(1, 7)$ , $- 6 x + y = - 1$

Step 1

Add $6 x$ to both sides of the equation.

$y = - 1 + 6 x$

Step 2

Find the slope when $y = - 1 + 6 x$ .

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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 $- 1$ and $6 x$ .

$y = 6 x - 1$

Step 2.2

Using the slope-intercept form, the slope is $6$ .

$m = 6$

Step 3

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

$m_{perpendicular} = - \frac{1}{6}$

Step 4

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

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Step 4.1

Use the slope $- \frac{1}{6}$ and a given point $(1, 7)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (7) = - \frac{1}{6} \cdot (x - (1))$

Step 4.2

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

$y - 7 = - \frac{1}{6} \cdot (x - 1)$

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 $- \frac{1}{6} \cdot (x - 1)$ .

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Step 5.1.1.1

Rewrite.

$y - 7 = 0 + 0 - \frac{1}{6} \cdot (x - 1)$

Step 5.1.1.2

Simplify by adding zeros.

$y - 7 = - \frac{1}{6} \cdot (x - 1)$

Step 5.1.1.3

Apply the distributive property.

$y - 7 = - \frac{1}{6} x - \frac{1}{6} \cdot - 1$

Step 5.1.1.4

Combine $x$ and $\frac{1}{6}$ .

$y - 7 = - \frac{x}{6} - \frac{1}{6} \cdot - 1$

Step 5.1.1.5

Multiply $- \frac{1}{6} \cdot - 1$ .

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Step 5.1.1.5.1

Multiply $- 1$ by $- 1$ .

$y - 7 = - \frac{x}{6} + 1 (\frac{1}{6})$

Step 5.1.1.5.2

Multiply $\frac{1}{6}$ by $1$ .

$y - 7 = - \frac{x}{6} + \frac{1}{6}$

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 $7$ to both sides of the equation.

$y = - \frac{x}{6} + \frac{1}{6} + 7$

Step 5.1.2.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$y = - \frac{x}{6} + \frac{1}{6} + 7 \cdot \frac{6}{6}$

Step 5.1.2.3

Combine $7$ and $\frac{6}{6}$ .

$y = - \frac{x}{6} + \frac{1}{6} + \frac{7 \cdot 6}{6}$

Step 5.1.2.4

Combine the numerators over the common denominator.

$y = - \frac{x}{6} + \frac{1 + 7 \cdot 6}{6}$

Step 5.1.2.5

Simplify the numerator.

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Step 5.1.2.5.1

Multiply $7$ by $6$ .

$y = - \frac{x}{6} + \frac{1 + 42}{6}$

Step 5.1.2.5.2

Add $1$ and $42$ .

$y = - \frac{x}{6} + \frac{43}{6}$

Step 5.2

Reorder terms.

$y = - (\frac{1}{6} x) + \frac{43}{6}$

Step 5.3

Remove parentheses.

$y = - \frac{1}{6} x + \frac{43}{6}$

Step 6