Let's solve this step-by-step:
1. Identify the Slope of Line QR:
The given equation of line QR is:
[tex]\[
y = \frac{-1}{2}x + 1
\][/tex]
This is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. Here, the slope [tex]\( m \)[/tex] of line QR is [tex]\( \frac{-1}{2} \)[/tex].
2. Find the Slope of the Perpendicular Line:
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. The negative reciprocal of [tex]\( \frac{-1}{2} \)[/tex] is [tex]\( 2 \)[/tex]. Therefore, the slope of the perpendicular line is [tex]\( 2 \)[/tex].
3. Use the Point-Slope Form to Find the Y-Intercept:
We need to find the equation of the line with the slope [tex]\( 2 \)[/tex] that passes through the point (5, 6). The equation of a line in slope-intercept form is:
[tex]\[
y = mx + b
\][/tex]
We know [tex]\( m = 2 \)[/tex], so we substitute [tex]\( m \)[/tex], [tex]\( x = 5 \)[/tex], and [tex]\( y = 6 \)[/tex] into the equation to find [tex]\( b \)[/tex]:
[tex]\[
6 = 2 \cdot 5 + b
\][/tex]
Simplifying this:
[tex]\[
6 = 10 + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
b = 6 - 10
\][/tex]
[tex]\[
b = -4
\][/tex]
4. Write the Final Equation:
Now that we have the slope and the y-intercept, we can write the equation of the line in slope-intercept form:
[tex]\[
y = 2x - 4
\][/tex]
Thus, the equation of the line perpendicular to line QR and passing through the point (5, 6) is:
[tex]\[
\boxed{y = 2x - 4}
\][/tex]
