Question

How do you write an equation of a line passing through (4, 2), perpendicular to `y=2x+3`?

Answer 1

See the entire solution process below:

Explanation:

The equation in the problem is in slope-intercept form. The slope-intercept form of a linear equation is: `y = color(red)(m)x + color(blue)(b)`

Where `color(red)(m)` is the slope and `color(blue)(b)` is the y-intercept value.

`y = color(red)(2)x + color(blue)(3)`

Therefore the slope of this line is `m = 2`

Let us call the slope of the perpendicular line `m_p`. By definition, the slope of a perpendicular line is:

`m_p = -1/m`

Therefore, for this problem, the slope of the perpendicular line is:

`m_p = -1/2`

We can use the slope-intercept formula, substitute the values from the points for `x` and `y` and substitute the slope we determined and solve for `b`:

`2 = (color(red)(-1/2) * 4) + color(blue)(b)`

`2 = -2 + color(blue)(b)`

`color(red)(2) + 2 = color(red)(2) - 2 + color(blue)(b)`

`4 = 0 + color(blue)(b)`

`4 = color(blue)(b)`

Which means the equation is:

`y = color(red)(-1/2)x + color(blue)(4)`

Or, we could use the point-slope formula to also write and equation for the line. The point-slope formula states: `(y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))`

Where `color(blue)(m)` is the slope and `color(red)(((x_1, y_1)))` is a point the line passes through.

Substituting the slope we calculated and the values from the point in the problem gives:

`(y - color(red)(2)) = color(blue)(-1/2)(x - color(red)(4))`