1. What Is The Equation Of A Line Parallel To Y = 4/5x + 2 That Passes Through (1, 2)?2. What Is The
Understanding how to find the equation of a line parallel to a given line and passing through a specific point is a fundamental concept in algebra and coordinate geometry. This tutorial explores the step-by-step process of determining such a line, focusing on the line parallel to Y = 4/5x + 2 that passes through the point (1, 2). Whether you're a student preparing for exams or someone interested in grasping the principles of linear equations, this comprehensive guide will clarify the process and provide useful tips to master similar problems.
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Understanding the Basics: What Is a Parallel Line?
Before diving into calculations, it's essential to understand what parallel lines are and their properties.
Definition of Parallel Lines
- Parallel lines are two or more lines in a plane that never intersect.
- They maintain a constant distance from each other.
- Parallel lines share the same slope, but have different y-intercepts.
Why Is Slope Important?
- The slope indicates the steepness or incline of a line.
- Lines with identical slopes are parallel.
Analyzing the Given Line: Y = 4/5x + 2
The given line is in slope-intercept form, which is:
\[ y = mx + b \]
Where:
- \( m \) is the slope
- \( b \) is the y-intercept
For the line \( y = \frac{4}{5}x + 2 \):
- Slope (m): \( \frac{4}{5} \)
- Y-intercept (b): 2
Since parallel lines share the same slope, any line parallel to this line must also have a slope of \( \frac{4}{5} \).
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Finding the Equation of a Parallel Line Passing Through (1, 2)
The key steps involve:
- Recognizing the slope of the new line.
- Using the point-slope form with the given point.
- Converting the equation to slope-intercept form if needed.
Step 1: Identify the Slope
- The slope \( m \) of the new line is the same as the original:
\[ m = \frac{4}{5} \]
Step 2: Apply the Point-Slope Formula
The point-slope form of a line's equation is:\[ y - y1 = m (x - x1) \]
Where:
- \( (x1, y1) \) is the point the line passes through
- \( m \) is the slope
Plugging in the point \( (1, 2) \):
\[ y - 2 = \frac{4}{5}(x - 1) \]
Step 3: Simplify to Slope-Intercept Form
Distribute the slope:\[ y - 2 = \frac{4}{5}x - \frac{4}{5} \]
Add 2 to both sides:
\[ y = \frac{4}{5}x - \frac{4}{5} + 2 \]
Express 2 as a fraction with denominator 5:
\[ 2 = \frac{10}{5} \]
So:
\[ y = \frac{4}{5}x - \frac{4}{5} + \frac{10}{5} \]
Combine the constants:
\[ y = \frac{4}{5}x + \frac{6}{5} \]
Final Equation:
\[ y = \frac{4}{5}x + \frac{6}{5} \]
This is the equation of the line parallel to \( y = \frac{4}{5}x + 2 \) that passes through the point \( (1, 2) \).
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Key Points to Remember
When solving similar problems, keep these points in mind:
- Parallel lines share the same slope.
- The point-slope formula is a powerful tool for deriving line equations passing through a specific point.
- Always convert fractions to simplest form for clarity.
- You can rearrange the equation into slope-intercept form for easier interpretation.
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Additional Examples and Variations
To deepen your understanding, here are some variations of the problem:
Example 1: Find the equation of a line parallel to \( y = -\frac{2}{3}x + 4 \) passing through \( (-2, 5) \).
Solution:
- Slope \( m = -\frac{2}{3} \)
- Point \( (-2, 5) \)
Applying point-slope form:
\[ y - 5 = -\frac{2}{3}(x + 2) \]
Distribute:
\[ y - 5 = -\frac{2}{3}x - \frac{4}{3} \]
Add 5 (or \( \frac{15}{3} \)) to both sides:
\[ y = -\frac{2}{3}x - \frac{4}{3} + \frac{15}{3} \]
Simplify:
\[ y = -\frac{2}{3}x + \frac{11}{3} \]
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Example 2: Find the equation of a line parallel to \( y = 3x - 7 \) passing through \( (0, 5) \).
Solution:
- Slope \( m = 3 \)
- Point \( (0, 5) \)
Equation:
\[ y - 5 = 3(x - 0) \]
\[ y - 5 = 3x \]
\[ y = 3x + 5 \]
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Applications of Parallel Line Equations
Understanding how to find the equation of parallel lines has practical applications in various fields:
- Architecture: Designing structures with parallel walls or beams.
- Engineering: Ensuring components maintain consistent spacing.
- Navigation: Plotting routes that are parallel to existing paths.
- Data Analysis: Fitting parallel lines to model trends.
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SEO Optimization Tips for Linear Equation Articles
To ensure your content reaches the right audience, consider the following SEO strategies:
- Use relevant keywords like "equation of a line," "parallel lines," "point-slope form," and "linear equations."
- Incorporate headings and subheadings with descriptive titles.
- Include step-by-step examples and explanations.
- Use bullet points and numbered lists for clarity.
- Optimize meta descriptions and image alt text if including visuals.
- Link to related topics such as "slope-intercept form," "point-slope form," or "linear equations."
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Conclusion
Finding the equation of a line parallel to a given line and passing through a specific point is a foundational skill in algebra. By understanding the properties of parallel lines, leveraging the slope, and applying the point-slope form, you can quickly derive the required equations. For instance, for the line parallel to \( y = \frac{4}{5}x + 2 \) passing through \( (1, 2) \), the resulting equation is \( y = \frac{4}{5}x + \frac{6}{5} \). Mastery of these techniques enhances your problem-solving toolkit and prepares you for more complex geometric and algebraic challenges.
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Remember: Practice with different slopes and points to solidify your understanding, and always double-check your calculations for accuracy.
