# The line segment joining the points A (x ; y) and B (3 ;-5) is perpendicular to CD, whose gradient is 3/4 . Express x in terms of y.

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### 1 Answer

You should use the equation that relates the gradients (slopes) of two perpendicular segments, such that:

`m_(AB)*m_(CD) = -1`

You may evaluate the gradient of the segment AB using the following formula, such that:

`m_(AB) = (y_B - y_A)/(x_B - x_A)`

The problem provides the coordinates of the points A and B, hence, you may replace the values in equation above, such that:

`m_(AB) = (-5 - y)/(3 - x)`

Replacing `(-5 - y)/(3 - x)` for `m_(AB)` and `3/4` for `m_(CD)` yields:

`(-5 - y)/(3 - x)*(3/4) = -1 => (-5 - y)/(3 - x) = -1/(3/4)`

`(-5 - y)/(3 - x) = -4/3 => -4(3 - x) = 3(-5 - y)`

`-12 + 4x = -15 - 3y`

Since you need to express x in terms of y, you need to isolate the term that contains x to the left side, such that:

`4x = -3y - 15 + 12 => 4x = -3y - 3 => x = (-3/4)y - 3/4`

`x = (-3/4)(y + 1)`

**Hence, expressing x in terms of y, under the given conditions, yields **`x = (-3/4)(y + 1).`