Algebra Examples

$x - y = 5$ and $x + y = - 10$

Step 1

Find the slope and y-intercept of the first equation.

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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

Subtract $x$ from both sides of the equation.

$- y = 5 - x$

Step 1.1.3

Divide each term in $- y = 5 - x$ by $- 1$ and simplify.

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Step 1.1.3.1

Divide each term in $- y = 5 - x$ by $- 1$ .

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

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Dividing two negative values results in a positive value.

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

Step 1.1.3.2.2

Divide $y$ by $1$ .

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

Step 1.1.3.3

Simplify the right side.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Divide $5$ by $- 1$ .

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

Step 1.1.3.3.1.2

Dividing two negative values results in a positive value.

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

Step 1.1.3.3.1.3

Divide $x$ by $1$ .

$y = - 5 + x$

Step 1.1.4

Reorder $- 5$ and $x$ .

$y = x - 5$

Step 1.2

Find the values of $m$ and $b$ using the form $y = m x + b$ .

$m_{1} = 1$

$b = - 5$

$m_{1} = 1$

$b = - 5$

Step 2

Find the slope and y-intercept of the second equation.

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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

Subtract $x$ from both sides of the equation.

$y = - 10 - x$

Step 2.1.3

Reorder $- 10$ and $- x$ .

$y = - x - 10$

Step 2.2

Find the values of $m$ and $b$ using the form $y = m x + b$ .

$m_{2} = - 1$

$b = - 10$

$m_{2} = - 1$

$b = - 10$

Step 3

Compare the slopes $m$ of the two equations.

$m_{1} = 1, m_{2} = - 1$

Step 4

Compare the decimal form of one slope with the negative reciprocal of the other slope. If they are equal, then the lines are perpendicular. If the they are not equal, then the lines are not perpendicular.

$m_{1} = 1, m_{2} = 1$

Step 5

The equations are perpendicular because the slopes of the two lines are negative reciprocals.

Perpendicular

Step 6