Determine if Perpendicular Are the lines defined by the equations 5y+x=10 and y=5x+4 perpendicular?

Are the lines defined by the equations $5 y + x = 10$ and $y = 5 x + 4$ perpendicular?

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

Rewrite in slope-intercept form.

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

$y = m x + b$

Subtract $x$ from both sides of the equation.

$5 y = 10 - x$

Divide each term in $5 y = 10 - x$ by $5$ and simplify.

Divide each term in $5 y = 10 - x$ by $5$ .

$\frac{5 y}{5} = \frac{10}{5} + \frac{- x}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 y}{5} = \frac{10}{5} + \frac{- x}{5}$

Divide $y$ by $1$ .

$y = \frac{10}{5} + \frac{- x}{5}$

Simplify the right side.

Simplify each term.

Divide $10$ by $5$ .

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

Move the negative in front of the fraction.

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

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

Reorder $2$ and $- \frac{x}{5}$ .

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

Reorder terms.

$y = - (\frac{1}{5} x) + 2$

Remove parentheses.

$y = - \frac{1}{5} x + 2$

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

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

$b = 2$

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

$b = 2$

Use the slope-intercept form to find the slope and y-intercept.

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

$y = m x + b$

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

$m_{2} = 5$

$b = 4$

$m_{2} = 5$

$b = 4$

Compare the slopes $m$ of the two equations.

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

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.

$m_{1} = - 0.2, m_{2} = - 0.2$

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

Perpendicular

Determine if Perpendicular Are the lines defined by the equations 5y+x=10 and y=5x+4 perpendicular?

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