al441280
Answered

Westonci.ca is the premier destination for reliable answers to your questions, brought to you by a community of experts. Get immediate and reliable answers to your questions from a community of experienced experts on our platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.

What is the equation, in slope-intercept form, of the line that is perpendicular to the line [tex]\( y - 4 = -\frac{2}{3}(x - 6) \)[/tex] and passes through the point [tex]\((-2, -2)\)[/tex]?

A. [tex]\( y = -\frac{2}{3}x - \frac{10}{3} \)[/tex]
B. [tex]\( y = -\frac{2}{3}x + \frac{10}{3} \)[/tex]
C. [tex]\( y = \frac{3}{2}x - 1 \)[/tex]
D. [tex]\( y = \frac{3}{2}x + 1 \)[/tex]

Sagot :

GinnyAnswer
To find the equation of the line that is perpendicular to the line [tex]\( y - 4 = -\frac{2}{3}(x - 6) \)[/tex] and passes through the point [tex]\((-2, -2)\)[/tex], we need to follow these steps:

1. Determine the slope of the given line:

The equation is in point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( m \)[/tex] is the slope. For the given line [tex]\( y - 4 = -\frac{2}{3}(x - 6) \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\(-\frac{2}{3}\)[/tex].

2. Find the slope of the perpendicular line:

The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. Therefore, the perpendicular slope [tex]\( m_{\perp} \)[/tex] will be:
[tex]\[ m_{\perp} = -\frac{1}{-\frac{2}{3}} = \frac{3}{2} \][/tex]

3. Use the point-slope form to determine the equation of the new line:

We need the equation of the line that passes through the point [tex]\((-2, -2)\)[/tex] with slope [tex]\( \frac{3}{2} \)[/tex]. The point-slope form is:
[tex]\[ y - y_1 = m_{\perp}(x - x_1) \][/tex]
Plugging in the point [tex]\((-2, -2)\)[/tex] and the slope [tex]\( \frac{3}{2} \)[/tex], we get:
[tex]\[ y - (-2) = \frac{3}{2}(x - (-2)) \][/tex]
Simplify this:
[tex]\[ y + 2 = \frac{3}{2}(x + 2) \][/tex]

4. Convert to slope-intercept form [tex]\( y = mx + b \)[/tex]:

Distribute [tex]\( \frac{3}{2} \)[/tex] on the right-hand side:
[tex]\[ y + 2 = \frac{3}{2}x + \frac{3}{2} \cdot 2 \][/tex]
[tex]\[ y + 2 = \frac{3}{2}x + 3 \][/tex]

Subtract 2 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = \frac{3}{2}x + 3 - 2 \][/tex]
[tex]\[ y = \frac{3}{2}x + 1 \][/tex]

So, the equation of the line in slope-intercept form that is perpendicular to the given line and passes through the point [tex]\((-2, -2)\)[/tex] is:

[tex]\[ y = \frac{3}{2}x + 1 \][/tex]

The correct choice is:
[tex]\[ \boxed{y = \frac{3}{2} x + 1} \][/tex]
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.