Question

For each of the following, state the equation of a perpendicular line that passes through (0, 0). Then using the slope of the new equation, find x if the point P(x, 4) lies on the new line. y=3x-1 y=1/4 x+2

Answer 1

The answer is below

Explanation:

a) y=3x-1

The standard equation of a line is given by:

y = mx + c

Where m is the slope of the line and c is the intercept on the y axis.

Given that y=3x-1, comparing with the standard equation of a line, the slope (m) = 3, Two lines with slope a and b are perpendicular if the product of their slope is -1 i.e. ab = -1. Let the line perpendicular to y=3x-1 be d, to get the slope of the perpendicular line, we use:

3 × d = -1

d = -1/3

To find the equation of the perpendicular line passing through (0,0), we use:

`y-y_1=d(x-x_1)\\d\ is\ the \ slope:\\y-0=-(1)/(3) (x-0)\\y=-(1)/(3)x`

To find x if the point P(x, 4) lies on the new line, insert y = 4 and find x:

`y=-(1)/(3)x\\ 4=-(1)/(3)x\\-x=12\\x=-12`

b) y=1/4 x+2

Given that y=1/4 x+2, comparing with the standard equation of a line, the slope (m) = 1/4. Let the line perpendicular to y=1/4 x+2 be f, to get the slope of the perpendicular line, we use:

1/4 × f = -1

f = -4

To find the equation of the perpendicular line passing through (0,0), we use:

`y-y_1=f(x-x_1)\\f\ is\ the \ slope:\\y-0=-4 (x-0)\\y=-4x`

To find x if the point P(x, 4) lies on the new line, insert y = 4 and find x:

`y=-4}x\\ 4=-4x\\x=-1`