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23. Given the line [tex] y = -\frac{3}{4} x + 1 [/tex], find the equation of a line perpendicular to that line that passes through the point [tex] (0,6) [/tex].

A. [tex] y = \frac{3}{4} x + 6 [/tex]
B. [tex] y = -\frac{3}{4} x + 6 [/tex]
C. [tex] y = -\frac{4}{3} x + 6 [/tex]
D. [tex] y = \frac{4}{3} x + 6 [/tex]


Sagot :

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]