Find the line that is perpendicular to y=x+2

Question

and passes through the point (3,0)
.(1 point)
Responses

y=−2x+6
y is equal to negative 2 x plus 6

y=−x+3
y is equal to negative x plus 3

y=12x−32
y is equal to 1 half x minus 3 halves

y=x−3

Answer 1

To find the equation of the line that is perpendicular to \( y = x + 2 \) and passes through the point \( (3, 0) \), follow these steps:

1. Determine the slope of the given line.
   The given line \( y = x + 2 \) has a slope of \( 1 \) (the coefficient of \( x \)).

2. Find the slope of the perpendicular line.
   The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Since the slope of the original line is \( 1 \), the slope of the perpendicular line will be: \[ m = -\frac{1}{1} = -1 \]

3. Use the point-slope form of the line.
   We can use the point-slope form \( y - y_1 = m(x - x_1) \):

   • Slope \( m = -1 \)
   • Point \( (x_1, y_1) = (3, 0) \) Plugging in the values: \[ y - 0 = -1(x - 3) \] Simplifying this: \[ y = -x + 3 \]

So, the equation of the line that is perpendicular to \( y = x + 2 \) and passes through the point \( (3, 0) \) is: \[ y = -x + 3 \]

This corresponds to the second response:

• y = −x + 3