Answered

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3. Choose the correct answer.

A given line has the equation [tex]3x - 5y = 20[/tex]. Which of the following equations represents the line perpendicular to the given line and passes through the point [tex](-6, 4)[/tex]?

A. [tex]y = -\frac{3}{5}x + 0.4[/tex]
B. [tex]y = -\frac{5}{3}x + 10[/tex]
C. [tex]y = -\frac{5}{3}x - 6[/tex]
D. [tex]5x - 3y = -18[/tex]


Sagot :

To solve the problem of finding the equation of a line that is perpendicular to the given line [tex]\(3x - 5y = 20\)[/tex] and passes through the point [tex]\((-6, 4)\)[/tex], let's follow step-by-step instructions:

1. Convert the given line equation to slope-intercept form:

The given line equation is:
[tex]\[ 3x - 5y = 20 \][/tex]

To convert this to slope-intercept form ([tex]\(y = mx + b\)[/tex]), solve for [tex]\(y\)[/tex]:
[tex]\[ -5y = -3x + 20 \][/tex]
Divide by [tex]\(-5\)[/tex]:
[tex]\[ y = \frac{3}{5}x - 4 \][/tex]

Therefore, the corresponding slope [tex]\(m\)[/tex] of the given line is:
[tex]\[ m = \frac{3}{5} \][/tex]

2. Find the slope of the line perpendicular to the given line:

Perpendicular slopes are negative reciprocals of each other; therefore, if the slope of the given line is [tex]\(\frac{3}{5}\)[/tex], the slope [tex]\(m'\)[/tex] of the perpendicular line will be:
[tex]\[ m' = -\frac{1}{m} = -\frac{1}{\frac{3}{5}} = -\frac{5}{3} \][/tex]

3. Use the point-slope form of the equation of a line that passes through the point [tex]\((-6, 4)\)[/tex]:

The point-slope form of a line is given by:
[tex]\[ y - y_1 = m'(x - x_1) \][/tex]

Where [tex]\((x_1, y_1)\)[/tex] is [tex]\((-6, 4)\)[/tex], and [tex]\(m' = -\frac{5}{3}\)[/tex]:
[tex]\[ y - 4 = -\frac{5}{3} (x + 6) \][/tex]

4. Simplify the equation to slope-intercept form (if necessary):

Distribute [tex]\(-\frac{5}{3}\)[/tex]:
[tex]\[ y - 4 = -\frac{5}{3} x - 10 \][/tex]

Add [tex]\(4\)[/tex] to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[ y = -\frac{5}{3}x - 6 \][/tex]

5. Choose the correct answer:

From the given options:
[tex]\[ y = -\frac{5}{3}x - 6 \][/tex]

Matches exactly with the equation we derived. Thus, the correct answer is:

[tex]\[ y = -\frac{5}{3}x - 6 \][/tex]

Therefore, the correct choice is:
[tex]\[ \boxed{3} \][/tex]