To find the equation of a line that is perpendicular to the line QR and passes through the point (5, 6), we need to follow these steps:
1. Determine the slope of line QR:
First, rewrite the equation of line QR [tex]\(x + 2y = 2\)[/tex] in slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
[tex]\[
x + 2y = 2
\][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[
2y = -x + 2
\][/tex]
[tex]\[
y = -\frac{1}{2}x + 1
\][/tex]
So, the slope of line QR ([tex]\(m_{\text{QR}}\)[/tex]) is [tex]\(-\frac{1}{2}\)[/tex].
2. Find the slope of the perpendicular line:
Lines that are perpendicular have slopes that are negative reciprocals of each other. If the slope of line QR is [tex]\(-\frac{1}{2}\)[/tex], the slope of the line perpendicular to it ([tex]\(m_{\perp}\)[/tex]) will be:
[tex]\[
m_{\perp} = -\frac{1}{-\frac{1}{2}} = 2
\][/tex]
3. Find the y-intercept of the perpendicular line:
To find the y-intercept ([tex]\(b\)[/tex]), we use the point [tex]\((5, 6)\)[/tex] that lies on the line. The equation of the perpendicular line in slope-intercept form is [tex]\(y = mx + b\)[/tex], where [tex]\(m = 2\)[/tex].
Plug in the point [tex]\((5, 6)\)[/tex]:
[tex]\[
6 = 2(5) + b
\][/tex]
[tex]\[
6 = 10 + b
\][/tex]
[tex]\[
b = 6 - 10
\][/tex]
[tex]\[
b = -4
\][/tex]
4. Write the equation of the perpendicular line:
Now that we have the slope [tex]\(m = 2\)[/tex] and the y-intercept [tex]\(b = -4\)[/tex], the equation of the line in slope-intercept form is:
[tex]\[
y = 2x - 4
\][/tex]
Thus, the equation of the line perpendicular to line QR and passing through the point (5, 6) is [tex]\(y = 2x - 4\)[/tex].
The correct answer is:
[tex]\[
y = 2x - 4
\][/tex]
