Question

What is the equation, in slope-intercept form, of line B?

Line A passes through point (−2, −1) and is perpendicular to the graph of y = −1/2x + 6.
Line B is parallel to line A and passes through the point (2, 1).

Answer 1

Line A Equation

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

Line B Equation:

`y = 2x - 3`

Explanation:

We are given a line whose equation is:

`y = -(1)/(2)x + 6 \cdots [1]`

The general slope-intercept form of a line equation is:

`y = mx + b`

where
`m` is the slope and
`b` is the y-intercept

The line defined by equation [1] has a slope of
`-(1)/(2)` and a y-intercept of 6.

Lines which are parallel to each other will have the same slope,
`m`, but different y-intercepts,
`b`

Line A Equation

• We are given that line A is parallel to the given line. This means that line A has the same slope of
  `-(1)/(2)`
  
• We can therefore write down the equation of line A as:
  
  `y = -(1)/(2)x + b \cdots[A]`
  
• To compute
  `b`, we note that the line A passes through (-2, -1). These are values of x and y which satisfy equation [A}

• Plug in these values for
  `x` and
  `y` and solve for
  `b` to get the complete :
  `-1 = -(1)/(2)\cdot(-2) + b\\\\-1 = 1 + b\\\\b = -2\\\\`

• So the equation for line A is
  
  `y = -(1)/(2)x -2`

• One way to verify this is by actually graphing this equation using a graphing calculator and verifying it is indeed parallel to the given line and does pass through (-2, -1)

• The first image provides this visual proof

Line B Equation

• A line which is perpendicular to another line will have a slope equal to the negative of the reciprocal of the other line. Thus if one line has slope
  `m`, a perpendicular line will have slope
  `-(1)/(m)`
  

• Since the given line has a slope of
  `-(1)/(2)` , line B should have a slope whish is the negative of the reciprocal of this number.
  
  `\textrm{Reciprocal of } -(1)/(2) = -2}\\\\\textrm{Negative of -2 is 2}`
  

• So slope of line B is
  `2` and we can write the equation of line B as:
  `y = 2x + b \cdots[B]\\\\`
  

• We proceed to find
  `b` using the same technique as we did for line A:
  Line B passes through (2, 1)
  

• Plug these values into equation [B}:
  
  `1 = 2(2) + b\\\\1 = 4 + b\\\\b = -3`
  

• So equation of line B is
  
  `y = 2x - 3`
  

• Graph for line B is attached as a second image

Hope that helps