Question

Which equation represents a line which is perpendicular to the line y=-1/2x−6?

Answer 1

To find a perpendicular line to y = -1/2x - 6, the slope of the new line should be 2, which is the negative reciprocal of -1/2. Any line with the form y = 2x + b, where b is any real number, represents a line perpendicular to the given line.

Step-by-step explanation:

To find a line perpendicular to the given line y = -1/2x - 6, we need to determine the slope of the perpendicular line. The slope of the given line is -1/2. Since perpendicular lines have slopes that are negative reciprocals of each other, the slope of our perpendicular line should be 2 (the negative reciprocal of -1/2).

The general form of a line is y = mx + b, where m is the slope and b is the y-intercept. Inserting our slope into this formula gives us the equation of the perpendicular line as y = 2x + b, where b can be any real number depending on where the line crosses the y-axis.

Therefore, a line that is perpendicular to y = -1/2x - 6 will have the form y = 2x + b.

Answer 2

The equation that represents a line that is perpendicular to the given line is;

`y-2x=-1`

Step-by-step explanation:

Given the equation;

`y=-(1)/(2)x-6`

The slope of the given equation is;

`m_1=-(1)/(2)`

for two lines to be perpendicular, their slopes must be the negative inverse of each other.

`\begin{gathered} m_1\cdot m_2=-1 \\ m_2=-(1)/(m_1) \end{gathered}`

substituting the given slope;

`\begin{gathered} m_2=-(1)/(m_1)=-(1)/((-(1)/(2))) \\ m_2=-1*-(2)/(1) \\ m_2=2 \end{gathered}`

Therefore, the slope of the perpendicular line must be equal to 2.

From the given answer choices, let us find the equation with a slope of 2;

`\begin{gathered} x-2y=8 \\ y=(1)/(2)x-4 \\ m=(1)/(2) \end{gathered}`
`\begin{gathered} x+2y=4 \\ y=-(1)/(2)x+2 \\ m=-(1)/(2) \end{gathered}`
`\begin{gathered} 2x+y=6 \\ y=-2x+6 \\ m=-2 \end{gathered}`
`\begin{gathered} y-2x=-1 \\ y=2x-1 \\ m=2 \end{gathered}`

Therefore, the equation that represents a line that is perpendicular to the given line is;

`y-2x=-1`