Certainly! Let’s go through the solution step-by-step to find the equation of a line that is perpendicular to the given line and passes through the point [tex]\((0, 6)\)[/tex].
### Step-by-Step Solution
1. Identify the Slope of the Original Line:
The given equation of the line is [tex]\( y = -\frac{3}{4} x + 1 \)[/tex].
Here, the slope [tex]\( m \)[/tex] of the original line is [tex]\(-\frac{3}{4}\)[/tex].
2. Determine the Slope of the Perpendicular Line:
Lines that are perpendicular to each other have slopes that are negative reciprocals. The negative reciprocal of [tex]\(-\frac{3}{4}\)[/tex] is [tex]\(\frac{4}{3}\)[/tex].
3. Write the Slope-Intercept Form of the Perpendicular Line:
The slope-intercept form of a line is given by [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Using the slope from step 2, the equation of our perpendicular line is [tex]\( y = \frac{4}{3} x + b \)[/tex].
4. Use the Given Point to Find the Y-Intercept:
We know the line passes through the point [tex]\((0, 6)\)[/tex].
Substituting [tex]\( x = 0 \)[/tex] and [tex]\( y = 6 \)[/tex] into the equation [tex]\( y = \frac{4}{3} x + b \)[/tex]:
[tex]\[
6 = \frac{4}{3} \cdot 0 + b
\][/tex]
[tex]\[
6 = b
\][/tex]
Hence, the y-intercept [tex]\( b \)[/tex] is 6.
5. Form the Equation of the Perpendicular Line:
Substituting the slope and the y-intercept into the slope-intercept form, the equation of the perpendicular line is:
[tex]\[
y = \frac{4}{3} x + 6
\][/tex]
### Conclusion
Thus, the equation of the line that is perpendicular to [tex]\( y = -\frac{3}{4} x + 1 \)[/tex] and passes through the point [tex]\((0, 6)\)[/tex] is:
[tex]\[
y = \frac{4}{3} x + 6
\][/tex]
Hence, the correct answer is:
[tex]\[ y = \frac{4}{3} x + 6 \][/tex]
