Question

Line R has the equation: y=1/2x+8.
Line S is perpendicular to that.
What is the equation for line S?
Answer
S goes through (2,-7).
Answered by GPT-4o mini
To find the equation of line S, which is perpendicular to line R, we first need to determine the slope of line R.

The slope of line R, given by the equation \(y = \frac{1}{2}x + 8\), is \(\frac{1}{2}\).

The slope of a line that is perpendicular to another line can be found by taking the negative reciprocal of the original slope. Therefore, the slope of line S will be:

\[
m_S = -\frac{1}{\frac{1}{2}} = -2
\]

Now that we have the slope of line S, we can write its equation in the slope-intercept form \(y = mx + b\). However, we need a point through which line S passes to determine the value of \(b\) (the y-intercept).

If we assume line S goes through the same point where line R intersects the y-axis (which is the point \((0, 8)\)), we can substitute \(x = 0\) and \(y = 8\) into the equation of line S:

\[
8 = -2(0) + b
\]
\[
8 = b
\]

Thus, if line S passes through the point \((0, 8)\), the equation for line S is:

\[
y = -2x + 8
\]

If line S does not have to pass through a specific point, then the general equation of line S can simply be written as:

\[
y = -2x + b
\]

where \(b\) can be any real number depending on the particular point you want line S to pass through.
Answered by GPT-4o mini
To find the equation of line S that is perpendicular to line R and passes through the point \((2, -7)\), we already determined that the slope of line S is \(-2\).

Now we can use the point-slope form of the equation of a line, which is given by:

\[
y - y_1 = m(x - x_1)
\]

where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.

Substituting in the known values:
- \(m = -2\)
- \(x_1 = 2\)
- \(y_1 = -7\)

The equation becomes:

\[
y - (-7) = -2(x - 2)
\]

This simplifies to:

\[
y + 7 = -2(x - 2)
\]

Distributing \(-2\) on the right side:

\[
y + 7 = -2x + 4
\]

Now, subtracting \(7\) from both sides gives:

\[
y = -2x + 4 - 7
\]
\[
y = -2x - 3
\]

Thus, the equation for line S is:

\[
y = -2x - 3
\]