Solved on Nov 13, 2023

Determine if the lines $y=\frac{-5}{6}x+\frac{9}{4}$ and $y=\frac{6}{5}x+\frac{5}{2}$ are parallel, perpendicular, or neither.

STEP 1

Assumptions1. The equation of line $a$ is $y=\frac{-5}{6} x+\frac{9}{4}$ . The equation of line $b$ is $y=\frac{6}{5} x+\frac{5}{}$
3. Two lines are parallel if their slopes are equal4. Two lines are perpendicular if the product of their slopes is -1

STEP 2

First, we need to find the slopes of the two lines. The slope of a line in the form $y=mx+b$ is $m$ .
For line $a$ , the slope $m_a$ is $\frac{-5}{6}$ .
For line $b$ , the slope $m_b$ is $\frac{6}{5}$ .

STEP 3

Now, we need to check if the lines are parallel. Two lines are parallel if their slopes are equal.So, we check if $m_a = m_b$ .

STEP 4

Substitute the values of $m_a$ and $m_b$ into the equation from3.
$\frac{-}{6} = \frac{6}{}$

STEP 5

This equation is not true, so the lines are not parallel.

STEP 6

Next, we need to check if the lines are perpendicular. Two lines are perpendicular if the product of their slopes is -1.
So, we check if $m_a \cdot m_b = -1$ .

STEP 7

Substitute the values of $m_a$ and $m_b$ into the equation from6.
$\frac{-5}{6} \cdot \frac{6}{5} = -1$

STEP 8

implify the left side of the equation.
$-1 = -1$

STEP 9

This equation is true, so the lines are perpendicular.
The lines $a$ and $b$ are perpendicular.

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Determine if the lines $y=\frac{-5}{6}x+\frac{9}{4}$ and $y=\frac{6}{5}x+\frac{5}{2}$ are parallel, perpendicular, or neither.

STEP 1

Assumptions1. The equation of line $a$ is $y=\frac{-5}{6} x+\frac{9}{4}$ . The equation of line $b$ is $y=\frac{6}{5} x+\frac{5}{}$
3. Two lines are parallel if their slopes are equal4. Two lines are perpendicular if the product of their slopes is -1

STEP 2

First, we need to find the slopes of the two lines. The slope of a line in the form $y=mx+b$ is $m$ .
For line $a$ , the slope $m_a$ is $\frac{-5}{6}$ .
For line $b$ , the slope $m_b$ is $\frac{6}{5}$ .

STEP 3

Now, we need to check if the lines are parallel. Two lines are parallel if their slopes are equal.So, we check if $m_a = m_b$ .

STEP 4

Substitute the values of $m_a$ and $m_b$ into the equation from3.
$\frac{-}{6} = \frac{6}{}$

STEP 5

This equation is not true, so the lines are not parallel.

STEP 6

Next, we need to check if the lines are perpendicular. Two lines are perpendicular if the product of their slopes is -1.
So, we check if $m_a \cdot m_b = -1$ .

STEP 7

Substitute the values of $m_a$ and $m_b$ into the equation from6.
$\frac{-5}{6} \cdot \frac{6}{5} = -1$

STEP 8

implify the left side of the equation.
$-1 = -1$

STEP 9

This equation is true, so the lines are perpendicular.
The lines $a$ and $b$ are perpendicular.

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Determine if the lines y=−56x+94y=\frac{-5}{6}x+\frac{9}{4}y=6−5​x+49​ and y=65x+52y=\frac{6}{5}x+\frac{5}{2}y=56​x+25​ are parallel, perpendicular, or neither.

STEP 1

Assumptions1. The equation of line aaa is y=−56x+94y=\frac{-5}{6} x+\frac{9}{4}y=6−5​x+49​ . The equation of line bbb is y=65x+5y=\frac{6}{5} x+\frac{5}{}y=56​x+5​3. Two lines are parallel if their slopes are equal4. Two lines are perpendicular if the product of their slopes is -1

STEP 2

First, we need to find the slopes of the two lines. The slope of a line in the form y=mx+by=mx+by=mx+b is mmm.For line aaa, the slope mam_ama​ is −56\frac{-5}{6}6−5​.For line bbb, the slope mbm_bmb​ is 65\frac{6}{5}56​.

STEP 3

Now, we need to check if the lines are parallel. Two lines are parallel if their slopes are equal.So, we check if ma=mbm_a = m_bma​=mb​.

STEP 4

Substitute the values of mam_ama​ and mbm_bmb​ into the equation from3.−6=6\frac{-}{6} = \frac{6}{}6−​=6​

STEP 5

This equation is not true, so the lines are not parallel.

STEP 6

Next, we need to check if the lines are perpendicular. Two lines are perpendicular if the product of their slopes is -1.So, we check if ma⋅mb=−1m_a \cdot m_b = -1ma​⋅mb​=−1.

STEP 7

Substitute the values of mam_ama​ and mbm_bmb​ into the equation from6.−56⋅65=−1\frac{-5}{6} \cdot \frac{6}{5} = -16−5​⋅56​=−1

STEP 8

implify the left side of the equation.−1=−1-1 = -1−1=−1

STEP 9

This equation is true, so the lines are perpendicular.The lines aaa and bbb are perpendicular.

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Determine if the lines $y=\frac{-5}{6}x+\frac{9}{4}$ and $y=\frac{6}{5}x+\frac{5}{2}$ are parallel, perpendicular, or neither.

Assumptions1. The equation of line $a$ is $y=\frac{-5}{6} x+\frac{9}{4}$ . The equation of line $b$ is $y=\frac{6}{5} x+\frac{5}{}$
3. Two lines are parallel if their slopes are equal4. Two lines are perpendicular if the product of their slopes is -1

First, we need to find the slopes of the two lines. The slope of a line in the form $y=mx+b$ is $m$ .
For line $a$ , the slope $m_a$ is $\frac{-5}{6}$ .
For line $b$ , the slope $m_b$ is $\frac{6}{5}$ .

Now, we need to check if the lines are parallel. Two lines are parallel if their slopes are equal.So, we check if $m_a = m_b$ .

Substitute the values of $m_a$ and $m_b$ into the equation from3.
$\frac{-}{6} = \frac{6}{}$

Next, we need to check if the lines are perpendicular. Two lines are perpendicular if the product of their slopes is -1.
So, we check if $m_a \cdot m_b = -1$ .

Substitute the values of $m_a$ and $m_b$ into the equation from6.
$\frac{-5}{6} \cdot \frac{6}{5} = -1$

implify the left side of the equation.
$-1 = -1$

This equation is true, so the lines are perpendicular.
The lines $a$ and $b$ are perpendicular.