Welcome to Westonci.ca, your go-to destination for finding answers to all your questions. Join our expert community today! Ask your questions and receive accurate answers from professionals with extensive experience in various fields on our platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
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]
### 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]
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.