The equation of line QR is [tex]x + 2y = 2[/tex]. What is the equation of a line perpendicular to line QR in slope-intercept form that contains point (5,6)?

A. [tex]y = -\frac{1}{2} x + \frac{17}{2}[/tex]
B. [tex]y = 2x - 4[/tex]
C. [tex]y = -\frac{1}{2} x + \frac{7}{2}[/tex]
D. [tex]y = 2x + 16[/tex]



Answer :

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]