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Answer :
To find the equation of a line that is perpendicular to a given line and passes through a specific point, follow these steps:
1. Determine the slope of the given line:
The equation of the given line is [tex]\( y = \frac{3}{4}x + 1 \)[/tex].
The slope ([tex]\( m \)[/tex]) of this line is [tex]\( \frac{3}{4} \)[/tex].
2. Find the slope of the perpendicular line:
Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]. If [tex]\( m_1 \)[/tex] is the slope of the given line, the slope [tex]\( m_2 \)[/tex] of the line perpendicular to it is:
[tex]\[ m_2 = -\frac{1}{m_1} \][/tex]
Given [tex]\( m_1 = \frac{3}{4} \)[/tex], the slope of the perpendicular line [tex]\( m_2 \)[/tex] is:
[tex]\[ m_2 = -\frac{1}{\frac{3}{4}} = -\frac{4}{3} \][/tex]
3. Use the point-slope form to find the equation:
The point-slope form of a line's equation is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\( (x_1, y_1) \)[/tex] is the point through which the line passes. We are given the point [tex]\( (-5, 11) \)[/tex]. Plugging in the values, we get:
[tex]\[ y - 11 = -\frac{4}{3}(x - (-5)) \][/tex]
[tex]\[ y - 11 = -\frac{4}{3}(x + 5) \][/tex]
4. Simplify the equation:
Distribute the slope and simplify to get the equation in slope-intercept form:
[tex]\[ y - 11 = -\frac{4}{3}x - \frac{4}{3} \cdot 5 \][/tex]
[tex]\[ y - 11 = -\frac{4}{3}x - \frac{20}{3} \][/tex]
[tex]\[ y = -\frac{4}{3}x - \frac{20}{3} + 11 \][/tex]
Convert 11 to a fraction over 3 for easier addition:
[tex]\[ 11 = \frac{33}{3} \][/tex]
So, the equation becomes:
[tex]\[ y = -\frac{4}{3}x - \frac{20}{3} + \frac{33}{3} \][/tex]
[tex]\[ y = -\frac{4}{3}x + \frac{13}{3} \][/tex]
Thus, the equation of the line that is perpendicular to [tex]\( y = \frac{3}{4}x + 1 \)[/tex] and passes through the point [tex]\((-5, 11)\)[/tex] is:
[tex]\[ \boxed{y = -\frac{4}{3}x + \frac{13}{3}} \][/tex]
So, the correct answer is:
A. [tex]\( y = -\frac{4}{3}x + \frac{13}{3} \)[/tex]
1. Determine the slope of the given line:
The equation of the given line is [tex]\( y = \frac{3}{4}x + 1 \)[/tex].
The slope ([tex]\( m \)[/tex]) of this line is [tex]\( \frac{3}{4} \)[/tex].
2. Find the slope of the perpendicular line:
Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]. If [tex]\( m_1 \)[/tex] is the slope of the given line, the slope [tex]\( m_2 \)[/tex] of the line perpendicular to it is:
[tex]\[ m_2 = -\frac{1}{m_1} \][/tex]
Given [tex]\( m_1 = \frac{3}{4} \)[/tex], the slope of the perpendicular line [tex]\( m_2 \)[/tex] is:
[tex]\[ m_2 = -\frac{1}{\frac{3}{4}} = -\frac{4}{3} \][/tex]
3. Use the point-slope form to find the equation:
The point-slope form of a line's equation is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\( (x_1, y_1) \)[/tex] is the point through which the line passes. We are given the point [tex]\( (-5, 11) \)[/tex]. Plugging in the values, we get:
[tex]\[ y - 11 = -\frac{4}{3}(x - (-5)) \][/tex]
[tex]\[ y - 11 = -\frac{4}{3}(x + 5) \][/tex]
4. Simplify the equation:
Distribute the slope and simplify to get the equation in slope-intercept form:
[tex]\[ y - 11 = -\frac{4}{3}x - \frac{4}{3} \cdot 5 \][/tex]
[tex]\[ y - 11 = -\frac{4}{3}x - \frac{20}{3} \][/tex]
[tex]\[ y = -\frac{4}{3}x - \frac{20}{3} + 11 \][/tex]
Convert 11 to a fraction over 3 for easier addition:
[tex]\[ 11 = \frac{33}{3} \][/tex]
So, the equation becomes:
[tex]\[ y = -\frac{4}{3}x - \frac{20}{3} + \frac{33}{3} \][/tex]
[tex]\[ y = -\frac{4}{3}x + \frac{13}{3} \][/tex]
Thus, the equation of the line that is perpendicular to [tex]\( y = \frac{3}{4}x + 1 \)[/tex] and passes through the point [tex]\((-5, 11)\)[/tex] is:
[tex]\[ \boxed{y = -\frac{4}{3}x + \frac{13}{3}} \][/tex]
So, the correct answer is:
A. [tex]\( y = -\frac{4}{3}x + \frac{13}{3} \)[/tex]
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