What Is The Equation Of The Line That Is Parallel To The Given Line And Passes Through The Point (3,
Understanding the equation of a line in coordinate geometry is fundamental for solving numerous mathematical problems. When asked to find a line parallel to a given line that passes through a specific point, it involves concepts like slope, point-slope form, and the properties of parallel lines. This comprehensive guide will walk you through the process step-by-step, ensuring you grasp the key ideas and can perform these calculations confidently.
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Understanding Lines and Their Equations
What Is a Line in Coordinate Geometry?
A line in the coordinate plane is a straight one-dimensional figure that extends infinitely in both directions. Its position is determined by its slope and intercepts. The equation of a line provides a mathematical description of its location and orientation.Common Forms of a Line's Equation
- Slope-Intercept Form: \( y = mx + b \)
- Point-Slope Form: \( y - y1 = m(x - x1) \)
- Standard Form: \( Ax + By + C = 0 \)
- \( m \) is the slope of the line
- \( (x1, y1) \) is a point on the line
Key Concepts for Finding Parallel Lines
What Does It Mean for Lines to Be Parallel?
Two lines are parallel if they are in the same plane and never intersect. This occurs precisely when their slopes are equal but their y-intercepts are different (unless they are the same line).Properties of Parallel Lines
- Same slope \( m \)
- Different y-intercepts \( b \) (or different points)
- No intersection points
Why Is Slope Important?
The slope determines the steepness and direction of a line. When lines are parallel, maintaining the same slope is essential for their parallelism.---
Step-by-Step Guide to Find the Equation of a Parallel Line Passing Through a Point
Suppose we are given:
- A line with an equation (say, in slope-intercept form or any form)
- A point \( (x0, y0) \), in this case, \( (3, y) \), where \( y \) is specified or to be determined
Our goal: Find the equation of a line parallel to the given line that passes through \( (3, y) \).
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Step 1: Identify the Slope of the Given Line
- Convert the given line into slope-intercept form \( y = mx + b \) if necessary
- Extract the slope \( m \)
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Step 2: Recognize the Slope of the Parallel Line
Since parallel lines have identical slopes, the new line passing through \( (3, y) \) will also have slope \( m = \frac{2}{3} \).---
Step 3: Use the Point-Slope Form to Write the Equation
The point-slope form of a line is: \[ y - y1 = m(x - x1) \] where \( (x1, y1) \) is the point the line passes through, and \( m \) is the slope.Applying the Point \( (3, y) \):
\[
y - y_0 = \frac{2}{3}(x - 3)
\]
Note: Since \( y_0 \) is unspecified, the line's equation can be written generally as:
\[
y - y_0 = \frac{2}{3}(x - 3)
\]
which simplifies to:
\[
y = \frac{2}{3}x + \left( y_0 - 2 \right)
\]
If the specific \( y \)-coordinate of the point is provided, substitute it directly.
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Step 4: Write the Equation in Slope-Intercept or Standard Form
Depending on the requirement, you can convert the point-slope form into:- Slope-Intercept Form:
- Standard Form:
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Handling Different Types of Given Line Equations
The process remains similar regardless of the initial form of the given line. The key steps are to determine the slope and then apply the point-slope formula.
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Example 1: Given line in standard form
Suppose the given line is \( 4x + y = 7 \) and the point is \( (3, y) \).Step 1: Find the slope:
\[
4x + y = 7 \Rightarrow y = -4x + 7
\]
Slope: \( m = -4 \)
Step 2: Use the point \( (3, y_0) \):
\[
y - y_0 = -4(x - 3)
\]
Step 3: Write the equation:
\[
y = -4x + 12 + y_0
\]
If \( y_0 \) is specified, substitute to get the explicit equation.
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Special Cases and Additional Considerations
When the Point Is Not Fully Specified
If the point is only partially specified, such as \( (3, y) \), and \( y \) is unknown, the general form of the equation will include \( y \) as a parameter. Alternatively, if the point has a specific \( y \)-coordinate, the equation can be fully determined.Vertical and Horizontal Lines
- Vertical lines: Have undefined slope; equation is \( x = x_0 \).
- Horizontal lines: Have slope \( 0 \); equation is \( y = y_0 \).
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Practical Applications of Finding Parallel Lines
Understanding how to find the equation of a line parallel to a given line passing through a specific point has numerous applications, including:
- Designing roads and pathways that run parallel to existing structures
- Creating parallel programming in computer graphics
- Solving geometric problems involving parallelism in architecture
- Analyzing trends in data fitting and regression lines
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Summary of the Process
- Identify the slope \( m \): Convert the given line to slope-intercept form or directly determine the slope from standard form.
- Use the point and the slope: Apply the point-slope formula \( y - y1 = m(x - x1) \).
- Simplify: Convert to the desired form (slope-intercept or standard form).
- Verify: Ensure the new line passes through the given point and is parallel to the original line.
Conclusion
Finding the equation of a line parallel to a given line and passing through a specific point involves understanding the core principles of slopes and line equations. By accurately determining the slope of the original line and applying the point-slope form, you can derive the new line's equation efficiently. Mastering these steps enhances your problem-solving skills in coordinate geometry and opens doors to tackling more complex geometric and algebraic challenges.
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Additional Resources and Practice Problems
To solidify your understanding, consider practicing with the following:
- Given the line \( y = 3x + 2 \), find the equation of the line parallel passing through \( (4, 5) \).
- Given the line \( 5x - 2y = 8 \), find the parallel line passing through \( (1, -3) \).
- Determine the equation of the line parallel to \( y = -\frac{1}{2}x + 4 \) passing through \( (0, 0) \).
By practicing these problems, you'll build confidence in applying the concepts of parallel lines and line equations in various contexts.
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Remember: The key always lies in understanding the slope and the point through which the new line passes. When these are clear, finding the equation of a parallel line becomes a straightforward process.
