To find the equation of a line that passes through the point [tex]\( (4, 5) \)[/tex] and is perpendicular to the line given by the equation [tex]\( y = \left(-\frac{1}{B}\right)x + 17 \)[/tex], we need to follow these steps:
1. Determine the Slope of the Given Line:
The given line's equation is [tex]\( y = \left(-\frac{1}{B}\right)x + 17 \)[/tex]. The slope of this line is [tex]\( m_{\text{given}} = -\frac{1}{B} \)[/tex].
2. Find the Slope of the Perpendicular Line:
The slopes of two perpendicular lines are opposite reciprocals of each other. Therefore, the slope [tex]\( m_{\text{perpendicular}} \)[/tex] of the line we are looking for is:
[tex]\[ m_{\text{perpendicular}} = -\frac{1}{m_{\text{given}}} = -\frac{1}{-\frac{1}{B}} = B \][/tex]
3. Use the Point-Slope Form to Write the Equation:
We know the slope of the perpendicular line is [tex]\( B \)[/tex] and it passes through the point [tex]\( (4, 5) \)[/tex]. Using the point-slope form of the equation of a line, which is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is the given point [tex]\((4, 5)\)[/tex] and [tex]\( m \)[/tex] is the slope [tex]\( B \)[/tex], we get:
[tex]\[ y - 5 = B(x - 4) \][/tex]
4. Convert to Slope-Intercept Form:
Solve for [tex]\( y \)[/tex] to write the equation in slope-intercept form [tex]\( y = mx + c \)[/tex].
[tex]\[
y - 5 = B(x - 4)
\][/tex]
[tex]\[
y - 5 = Bx - 4B
\][/tex]
[tex]\[
y = Bx - 4B + 5
\][/tex]
5. Identify the Y-Intercept:
The y-intercept [tex]\( c \)[/tex] can be found directly from the slope-intercept form of the equation derived above:
[tex]\[
y = Bx + (5 - 4B)
\][/tex]
So, [tex]\( c = 5 - 4B \)[/tex].
6. Present the Final Equation:
The final equation of the line in slope-intercept form is:
[tex]\[
y = Bx + (5 - 4B)
\][/tex]
Given the specific numerical values derived:
- Slope [tex]\( B = 8 \)[/tex]
- Y-intercept [tex]\( c = -27 \)[/tex]
Thus, the final equation of the line is:
[tex]\[
y = 8x - 27
\][/tex]
