Question

HELP ASAP

Write an equation of line passing through (-20, 4) and perpendicular to the line 2x + 5y = 10

Answer 1

`y = \displaystyle (5)/(2)\, x + 54`.

(Equivalently,
`5\, x - 2\, y = 108`.)

Explanation:

Rewrite the equation of the line
`2\, x + 5\, y = 10` in the slope-intercept form
`y = m\, x + b` to find the slope
`m`.

`5\, y = -2\, x + 10`.

`\displaystyle y = -(2)/(5)\, x + 2`.

Therefore, the slope of the line
`2\, x + 5\, y = 10` is
`(-2 / 5)`.

Let
`m_1` and
`m_2` denote the slope of two lines on a cartesian plane. If these two lines are perpendicular to one another, then
`m_1 \cdot m_2 = (-1)`.

In this question, the slope of the first line was already found
`m_1 = (-2 / 5)`. Solve
`m_1 \cdot m_2 = (-1)` for the slope
`m_2` of the unknown line:

`\begin{aligned}m_2 &= (-1)/(m_1) = (-1)/(-2 / 5) = (5)/(2)\end{aligned}`.

The following information are thus available for the requested line:

• The slope of this line is
  `\displaystyle m = (5)/(2)`.
• This line contains the point
  `(x_0,\, y_0) = (-20,\, 4)`.

Hence, the point-slope form
`(y - y_0) = m\, (x - x_0)` would be a possible choice for the equation of this line:

`\displaystyle y - 4 = (5)/(2)\, (x - (-20))`.

Simplify to obtain the slope-intercept form equation of this line:

`\displaystyle y = (5)/(2)\, x + 54`.

Equivalently:

`5\, x - 2\, y = 108`.