Answer :
To determine the equation of a line that is perpendicular to the given line \( y = 4x + 5 \) and passes through the point \( (8, 3) \), we follow these steps:
1. Identify the Slope of the Given Line:
- The given line \( y = 4x + 5 \) is in the slope-intercept form \( y = mx + b \) where \( m \) is the slope.
- Here, the slope \( m = 4 \).
2. Calculate the Slope of the Perpendicular Line:
- The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line.
- Therefore, the slope of the perpendicular line is \( -\frac{1}{4} \).
3. Use the Point-Slope Form to Find the Equation:
- We have the slope \( m = -\frac{1}{4} \) and a point \( (8, 3) \) through which the perpendicular line passes.
- The point-slope form of the equation of a line is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line.
- Plug in the values: \( x_1 = 8 \) and \( y_1 = 3 \).
[tex]\[ y - 3 = -\frac{1}{4}(x - 8) \][/tex]
4. Simplify the Equation:
- Distribute the slope on the right-hand side:
[tex]\[ y - 3 = -\frac{1}{4}x + 2 \][/tex]
- Add 3 to both sides to solve for \( y \):
[tex]\[ y = -\frac{1}{4}x + 2 + 3 \][/tex]
- Combine the constants:
[tex]\[ y = -\frac{1}{4}x + 5 \][/tex]
Therefore, the equation of the line that is perpendicular to \( y = 4x + 5 \) and passes through the point \( (8, 3) \) is:
[tex]\[ y = -\frac{1}{4}x + 5 \][/tex]
From the given options, the correct answer is:
C. [tex]\( y = -\frac{1}{4}x + 5 \)[/tex]
1. Identify the Slope of the Given Line:
- The given line \( y = 4x + 5 \) is in the slope-intercept form \( y = mx + b \) where \( m \) is the slope.
- Here, the slope \( m = 4 \).
2. Calculate the Slope of the Perpendicular Line:
- The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line.
- Therefore, the slope of the perpendicular line is \( -\frac{1}{4} \).
3. Use the Point-Slope Form to Find the Equation:
- We have the slope \( m = -\frac{1}{4} \) and a point \( (8, 3) \) through which the perpendicular line passes.
- The point-slope form of the equation of a line is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line.
- Plug in the values: \( x_1 = 8 \) and \( y_1 = 3 \).
[tex]\[ y - 3 = -\frac{1}{4}(x - 8) \][/tex]
4. Simplify the Equation:
- Distribute the slope on the right-hand side:
[tex]\[ y - 3 = -\frac{1}{4}x + 2 \][/tex]
- Add 3 to both sides to solve for \( y \):
[tex]\[ y = -\frac{1}{4}x + 2 + 3 \][/tex]
- Combine the constants:
[tex]\[ y = -\frac{1}{4}x + 5 \][/tex]
Therefore, the equation of the line that is perpendicular to \( y = 4x + 5 \) and passes through the point \( (8, 3) \) is:
[tex]\[ y = -\frac{1}{4}x + 5 \][/tex]
From the given options, the correct answer is:
C. [tex]\( y = -\frac{1}{4}x + 5 \)[/tex]
