Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
To find the equation of the line that is perpendicular to the line [tex]\( y = \frac{3}{5}x + 10 \)[/tex] and passes through the point [tex]\((15, -5)\)[/tex], let's go through the necessary steps:
1. Determine the slope of the original line:
The given line has an equation [tex]\( y = \frac{3}{5}x + 10 \)[/tex]. The slope [tex]\( m \)[/tex] of this line is [tex]\( \frac{3}{5} \)[/tex].
2. Find the slope of the perpendicular line:
Lines that are perpendicular to each other have slopes that are negative reciprocals. Hence, the slope of the line perpendicular to [tex]\( \frac{3}{5} \)[/tex] is:
[tex]\[ m_{\text{perpendicular}} = -\frac{1}{\frac{3}{5}} = -\frac{5}{3} \][/tex]
3. Use the point-slope form of the equation of a line:
The point-slope form is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope. Here, we need to substitute the point [tex]\((15, -5)\)[/tex] and the slope [tex]\(-\frac{5}{3}\)[/tex].
Start with the point-slope form:
[tex]\[ y - (-5) = -\frac{5}{3}(x - 15) \][/tex]
Simplify inside the parentheses:
[tex]\[ y + 5 = -\frac{5}{3}(x - 15) \][/tex]
4. Distribute the slope [tex]\(-\frac{5}{3}\)[/tex]:
[tex]\[ y + 5 = -\frac{5}{3}x + \left(-\frac{5}{3}\right) \cdot 15 \][/tex]
Calculate the constant term:
[tex]\[ -\frac{5}{3} \cdot 15 = -25 \][/tex]
So the equation becomes:
[tex]\[ y + 5 = -\frac{5}{3}x - 25 \][/tex]
5. Solve for [tex]\( y \)[/tex] to get the equation in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y = -\frac{5}{3}x - 25 - 5 \][/tex]
Simplify the constant term:
[tex]\[ y = -\frac{5}{3}x - 30 \][/tex]
However, upon closer examination:
[tex]\[ y + 5 = -\frac{5}{3}x + 25 \][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{5}{3}x + 25 - 5 \][/tex]
[tex]\[ y = -\frac{5}{3}x + 20 \][/tex]
Therefore, the equation of the line that is perpendicular to [tex]\( y = \frac{3}{5}x + 10 \)[/tex] and passes through the point [tex]\((15, -5)\)[/tex] is:
[tex]\[ y = -\frac{5}{3}x + 20 \][/tex]
The correct answer among the given choices is:
[tex]\[ \boxed{y = -\frac{5}{3}x + 20} \][/tex]
Hence, the correct option is:
[tex]\[ \boxed{4 \text{ (Corresponding to: } y=-\frac{5}{3} x+20 \text{)}} \][/tex]
1. Determine the slope of the original line:
The given line has an equation [tex]\( y = \frac{3}{5}x + 10 \)[/tex]. The slope [tex]\( m \)[/tex] of this line is [tex]\( \frac{3}{5} \)[/tex].
2. Find the slope of the perpendicular line:
Lines that are perpendicular to each other have slopes that are negative reciprocals. Hence, the slope of the line perpendicular to [tex]\( \frac{3}{5} \)[/tex] is:
[tex]\[ m_{\text{perpendicular}} = -\frac{1}{\frac{3}{5}} = -\frac{5}{3} \][/tex]
3. Use the point-slope form of the equation of a line:
The point-slope form is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope. Here, we need to substitute the point [tex]\((15, -5)\)[/tex] and the slope [tex]\(-\frac{5}{3}\)[/tex].
Start with the point-slope form:
[tex]\[ y - (-5) = -\frac{5}{3}(x - 15) \][/tex]
Simplify inside the parentheses:
[tex]\[ y + 5 = -\frac{5}{3}(x - 15) \][/tex]
4. Distribute the slope [tex]\(-\frac{5}{3}\)[/tex]:
[tex]\[ y + 5 = -\frac{5}{3}x + \left(-\frac{5}{3}\right) \cdot 15 \][/tex]
Calculate the constant term:
[tex]\[ -\frac{5}{3} \cdot 15 = -25 \][/tex]
So the equation becomes:
[tex]\[ y + 5 = -\frac{5}{3}x - 25 \][/tex]
5. Solve for [tex]\( y \)[/tex] to get the equation in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y = -\frac{5}{3}x - 25 - 5 \][/tex]
Simplify the constant term:
[tex]\[ y = -\frac{5}{3}x - 30 \][/tex]
However, upon closer examination:
[tex]\[ y + 5 = -\frac{5}{3}x + 25 \][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{5}{3}x + 25 - 5 \][/tex]
[tex]\[ y = -\frac{5}{3}x + 20 \][/tex]
Therefore, the equation of the line that is perpendicular to [tex]\( y = \frac{3}{5}x + 10 \)[/tex] and passes through the point [tex]\((15, -5)\)[/tex] is:
[tex]\[ y = -\frac{5}{3}x + 20 \][/tex]
The correct answer among the given choices is:
[tex]\[ \boxed{y = -\frac{5}{3}x + 20} \][/tex]
Hence, the correct option is:
[tex]\[ \boxed{4 \text{ (Corresponding to: } y=-\frac{5}{3} x+20 \text{)}} \][/tex]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.