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Sagot :
Let's solve the given problem step by step.
### Finding the Equation of the Perpendicular Line
1. Identify the slope of the given line:
The equation of the given line is [tex]\( y = 4x - 9 \)[/tex]. The slope [tex]\( m_1 \)[/tex] of this line is 4.
2. Determine the slope of the perpendicular line:
The slope of the line perpendicular to the given line is the negative reciprocal of the slope of the given line. Therefore, the slope [tex]\( m_\perpendicular \)[/tex] is:
[tex]\[ m_\perpendicular = -\frac{1}{m_1} = -\frac{1}{4} \][/tex]
3. Use the point-slope form to find the equation of the perpendicular line:
We use the point [tex]\((-7, -4)\)[/tex] and the slope [tex]\(-\frac{1}{4}\)[/tex] to find the equation. The point-slope form of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Substituting the values, we get:
[tex]\[ y - (-4) = -\frac{1}{4}(x - (-7)) \][/tex]
Simplifying this equation:
[tex]\[ y + 4 = -\frac{1}{4}(x + 7) \][/tex]
[tex]\[ y + 4 = -\frac{1}{4}x - \frac{7}{4} \][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{1}{4}x - \frac{7}{4} - 4 \][/tex]
Converting [tex]\(-4\)[/tex] to a fraction with the same denominator:
[tex]\[ y = -\frac{1}{4}x - \frac{7}{4} - \frac{16}{4} \][/tex]
[tex]\[ y = -\frac{1}{4}x - \frac{23}{4} \][/tex]
Simplifying the constant term:
[tex]\[ y = -\frac{1}{4}x - 5.75 \][/tex]
Thus, the equation of the perpendicular line is:
[tex]\[ y = -\frac{1}{4}x - 5.75 \][/tex]
### Finding the Equation of the Parallel Line
1. Identify the slope of the given line:
Recall that the slope [tex]\( m_1 \)[/tex] of the given line [tex]\( y = 4x - 9 \)[/tex] is 4.
2. Determine the slope of the parallel line:
The slope of the line parallel to the given line is the same as the slope of the given line. Therefore, the slope [tex]\( m_\parallel \)[/tex] is:
[tex]\[ m_\parallel = 4 \][/tex]
3. Use the point-slope form to find the equation of the parallel line:
We use the point [tex]\((-7, -4)\)[/tex] and the slope [tex]\(4\)[/tex] to find the equation. The point-slope form of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Substituting the values, we get:
[tex]\[ y - (-4) = 4(x - (-7)) \][/tex]
Simplifying this equation:
[tex]\[ y + 4 = 4(x + 7) \][/tex]
[tex]\[ y + 4 = 4x + 28 \][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[ y = 4x + 28 - 4 \][/tex]
[tex]\[ y = 4x + 24 \][/tex]
Thus, the equation of the parallel line is:
[tex]\[ y = 4x + 24 \][/tex]
### Final Equations
- Equation of the perpendicular line: [tex]\[ y = -\frac{1}{4}x - 5.75 \][/tex]
- Equation of the parallel line: [tex]\[ y = 4x + 24 \][/tex]
### Finding the Equation of the Perpendicular Line
1. Identify the slope of the given line:
The equation of the given line is [tex]\( y = 4x - 9 \)[/tex]. The slope [tex]\( m_1 \)[/tex] of this line is 4.
2. Determine the slope of the perpendicular line:
The slope of the line perpendicular to the given line is the negative reciprocal of the slope of the given line. Therefore, the slope [tex]\( m_\perpendicular \)[/tex] is:
[tex]\[ m_\perpendicular = -\frac{1}{m_1} = -\frac{1}{4} \][/tex]
3. Use the point-slope form to find the equation of the perpendicular line:
We use the point [tex]\((-7, -4)\)[/tex] and the slope [tex]\(-\frac{1}{4}\)[/tex] to find the equation. The point-slope form of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Substituting the values, we get:
[tex]\[ y - (-4) = -\frac{1}{4}(x - (-7)) \][/tex]
Simplifying this equation:
[tex]\[ y + 4 = -\frac{1}{4}(x + 7) \][/tex]
[tex]\[ y + 4 = -\frac{1}{4}x - \frac{7}{4} \][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{1}{4}x - \frac{7}{4} - 4 \][/tex]
Converting [tex]\(-4\)[/tex] to a fraction with the same denominator:
[tex]\[ y = -\frac{1}{4}x - \frac{7}{4} - \frac{16}{4} \][/tex]
[tex]\[ y = -\frac{1}{4}x - \frac{23}{4} \][/tex]
Simplifying the constant term:
[tex]\[ y = -\frac{1}{4}x - 5.75 \][/tex]
Thus, the equation of the perpendicular line is:
[tex]\[ y = -\frac{1}{4}x - 5.75 \][/tex]
### Finding the Equation of the Parallel Line
1. Identify the slope of the given line:
Recall that the slope [tex]\( m_1 \)[/tex] of the given line [tex]\( y = 4x - 9 \)[/tex] is 4.
2. Determine the slope of the parallel line:
The slope of the line parallel to the given line is the same as the slope of the given line. Therefore, the slope [tex]\( m_\parallel \)[/tex] is:
[tex]\[ m_\parallel = 4 \][/tex]
3. Use the point-slope form to find the equation of the parallel line:
We use the point [tex]\((-7, -4)\)[/tex] and the slope [tex]\(4\)[/tex] to find the equation. The point-slope form of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Substituting the values, we get:
[tex]\[ y - (-4) = 4(x - (-7)) \][/tex]
Simplifying this equation:
[tex]\[ y + 4 = 4(x + 7) \][/tex]
[tex]\[ y + 4 = 4x + 28 \][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[ y = 4x + 28 - 4 \][/tex]
[tex]\[ y = 4x + 24 \][/tex]
Thus, the equation of the parallel line is:
[tex]\[ y = 4x + 24 \][/tex]
### Final Equations
- Equation of the perpendicular line: [tex]\[ y = -\frac{1}{4}x - 5.75 \][/tex]
- Equation of the parallel line: [tex]\[ y = 4x + 24 \][/tex]
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