Question

The equations of three lines are given belovr.Line 1: 5y = 3x + 2Line 2: y = (5/3) x - 1Line 3: 6x + 10y = 6For each pair of lines, determine whether they are parallel, perpendicular, or neither.Line 1 and Line 2: O Parallel PerpendicularNeither??Line 1 and Line 3: O Parallel Perpendicular NeitherLine 2 and Line 3: O ParallelPerpendicular Neither

Answer 1

We are given the lines

Line 1

`5y=3x+2`

Line 2

`y=(5)/(3)x-1`

Line 3

`6x+10y=6`

We want to determine the pair that are parallel, perpendicular or neither

Solution

In order to determine that, we need to obtain the slope of each lines

Note: (1) if the slopes are equal, then they are parallel;

(2) If the product of the slopes equal neagtive one (-1), then they are perpendicular;

(3) Otherwise, they are neither

Now, we will obtain the slope of the lines

We will write each of the equation in this form

`y=mx+c`

Thus, the slope will be the coefficient of x (i.e m)

For Line 1

`\begin{gathered} 5y=3x+2 \\ y=(3)/(5)x+(2)/(5) \\ \text{Therefore,} \\ m_1=(3)/(5) \end{gathered}`

For Line 2

`\begin{gathered} y=(5)/(3)x-1 \\ \text{thus,} \\ m_2=(5)/(3) \end{gathered}`

For Line 3

`\begin{gathered} 6x+10y=6 \\ 10y=-6x+6 \\ y=-(6)/(10)x+(6)/(10) \\ y=-(3)/(5)x+(3)/(5) \\ \text{Therefore,} \\ m_3=-(3)/(5) \end{gathered}`

Now, Let us consider Line 1 and Line 2

`\begin{gathered} m_1\\e m_2\text{ (Not parallel)} \\ m_1* m_2=(3)/(5)*(5)/(3)=1\\e-1\text{ (not perpendicular)} \end{gathered}`

Therefore, Line 1 and Line 2 are neither parallel nor perpendicular

For Line 1 and Line 3

`\begin{gathered} m_1\\e m_{3\text{ }}\text{ (not parallel)} \\ m_1* m_3=(3)/(5)*-(3)/(5)=-(9)/(25)\\e-1\text{ (not perpendicular)} \end{gathered}`

Therefore, Line 1 and Line 3 are neither parallel nor perpendicular

We consider finally Line 2 and Line 3

`\begin{gathered} m_2\\e m_{3\text{ }}\text{ (not parallel)} \\ m_2* m_3=(5)/(3)*-(3)/(5)=-1\text{ (perpendicular)} \end{gathered}`

Therefore, Line 2 and Line 3 are perpendicular