Write an equation (a) in slope-intercept form and (b) in standard form for the line passing through (1,6) and perpendicular to 2x + 5y = 1. a) The equation of the line in slope-intercept form is (Type your answer in slope-intercept form. Use integers or fractions for any numbers in the equation.)
Write an equation (a) in slope-intercept form and (b) in standard form for the line passing through (1,6) and perpendicular to 2x + 5y = 1. a) The equation of the line in slope-intercept form is (Type your answer in slope-intercept form. Use integers or fractions for any numbers in the equation.)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Problem Statement:**
Write an equation **(a)** in slope-intercept form and **(b)** in standard form for the line passing through (1, 6) and perpendicular to \( 2x + 5y = 1 \).
---
**Solution:**
**a)** The equation of the line in slope-intercept form is [ ________ ].
*(Type your answer in slope-intercept form. Use integers or fractions for any numbers in the equation.)*
---
**Explanation:**
For part **(a)** of the problem, we need to determine the equation of a line in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
1. **Identify the slope of the given line \(2x + 5y = 1\):**
- Rewrite the given equation in slope-intercept form (isolate \(y\)):
\[
2x + 5y = 1 \implies 5y = -2x + 1 \implies y = -\frac{2}{5}x + \frac{1}{5}
\]
- The slope \((m)\) of the given line is \(-\frac{2}{5}\).
2. **Determine the slope of the perpendicular line:**
- The slope of a line perpendicular to another is the negative reciprocal of the slope of the given line:
\[
m_{\text{perpendicular}} = -\left(-\frac{2}{5}\right)^{-1} = \frac{5}{2}
\]
3. **Form the equation of the line passing through the point (1,6) with the perpendicular slope \(\frac{5}{2}\):**
- Use the point-slope form of the line equation \(y - y_1 = m(x - x_1)\), where \(m = \frac{5}{2}\) and \((x_1, y_1) = (1, 6)\):
\[
y - 6 = \frac{5}{2}(x - 1)
\]
- Simplify to get the slope-intercept form:
\[
y - 6 = \frac{5}{2}x - \frac{5}{2} \implies y =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb04437b-3f7e-4496-ba03-3a7f713d70b1%2F05c8119b-99e1-48fa-98bc-bea1ae996b18%2F85nk3mb_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Write an equation **(a)** in slope-intercept form and **(b)** in standard form for the line passing through (1, 6) and perpendicular to \( 2x + 5y = 1 \).
---
**Solution:**
**a)** The equation of the line in slope-intercept form is [ ________ ].
*(Type your answer in slope-intercept form. Use integers or fractions for any numbers in the equation.)*
---
**Explanation:**
For part **(a)** of the problem, we need to determine the equation of a line in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
1. **Identify the slope of the given line \(2x + 5y = 1\):**
- Rewrite the given equation in slope-intercept form (isolate \(y\)):
\[
2x + 5y = 1 \implies 5y = -2x + 1 \implies y = -\frac{2}{5}x + \frac{1}{5}
\]
- The slope \((m)\) of the given line is \(-\frac{2}{5}\).
2. **Determine the slope of the perpendicular line:**
- The slope of a line perpendicular to another is the negative reciprocal of the slope of the given line:
\[
m_{\text{perpendicular}} = -\left(-\frac{2}{5}\right)^{-1} = \frac{5}{2}
\]
3. **Form the equation of the line passing through the point (1,6) with the perpendicular slope \(\frac{5}{2}\):**
- Use the point-slope form of the line equation \(y - y_1 = m(x - x_1)\), where \(m = \frac{5}{2}\) and \((x_1, y_1) = (1, 6)\):
\[
y - 6 = \frac{5}{2}(x - 1)
\]
- Simplify to get the slope-intercept form:
\[
y - 6 = \frac{5}{2}x - \frac{5}{2} \implies y =
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