What Is The Equation Of The Line That Passes Through The Point (5, 4) And Has A Slopeof 6/5 is a fundamental question in coordinate geometry that helps students and enthusiasts understand how to derive the linear equation of a line given specific conditions. Determining the equation of a line involves understanding the relationship between its slope and points through which it passes. In this article, we will explore in detail how to find the equation of such a line, the concepts involved, and practical applications to reinforce your understanding.
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Understanding the Basics: What Is a Line Equation?
Before diving into the specific problem, it’s essential to grasp the foundational concepts related to line equations.
What Is a Line in Coordinate Geometry?
A line in a two-dimensional plane is a straight one-dimensional figure that extends infinitely in both directions. It can be uniquely identified by its slope and a point through which it passes.Common Forms of a Line Equation
There are several ways to express the equation of a line, including:- Slope-Intercept Form: y = mx + b
- Point-Slope Form: y - y₁ = m(x - x₁)
- Standard Form: Ax + By = C
In this context, the slope-intercept form and point-slope form are most relevant because they directly incorporate the slope and specific points.
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Key Concepts for Finding the Equation of a Line
To find the equation of a line passing through a point with a known slope, you need to understand and apply the following key concepts:
1. Slope of a Line (m)
The slope measures how steep the line is, calculated as the ratio of the change in y to the change in x between two points on the line: m = Δy / Δx.2. Coordinates of a Point (x₁, y₁)
The specific point through which the line passes provides a fixed location in the plane.3. The Point-Slope Formula
This formula combines the slope and a point to define the line: \[ y - y1 = m(x - x1) \]---
Calculating the Equation of the Line Passing Through (5, 4) With a Slope of 6/5
Applying the key concepts, we can now determine the line’s equation step by step.
Step 1: Identify the Known Values
- Point: (x₁, y₁) = (5, 4)
- Slope: m = 6/5
Step 2: Use the Point-Slope Formula
Plugging these into the formula: \[ y - 4 = \frac{6}{5}(x - 5) \]Step 3: Simplify the Equation
Distribute the slope: \[ y - 4 = \frac{6}{5}x - \frac{6}{5} \times 5 \] \[ y - 4 = \frac{6}{5}x - 6 \]Add 4 to both sides to solve for y:
\[ y = \frac{6}{5}x - 6 + 4 \]
\[ y = \frac{6}{5}x - 2 \]
Final Equation in Slope-Intercept Form:
\[ y = \frac{6}{5}x - 2 \]
This is the equation of the line passing through (5, 4) with a slope of 6/5.
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Understanding the Slope-Intercept Form of the Line
The final equation:
\[ y = \frac{6}{5}x - 2 \]
provides valuable insights:
- Slope (m): 6/5, indicating that for every 5 units increase in x, y increases by 6 units.
- Y-intercept (b): -2, meaning the line crosses the y-axis at (0, -2).
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Additional Methods to Find the Equation of a Line
While the point-slope form is straightforward when the slope and a point are known, other methods can also be employed:
1. Using Two Points
If two points are known, the process involves:- Calculating the slope from the two points.
- Using the point-slope form with either point.
2. Standard Form Conversion
Converting the slope-intercept form to standard form: \[ y = \frac{6}{5}x - 2 \] Multiply both sides by 5 to clear the fraction: \[ 5y = 6x - 10 \] Rearranged as: \[ 6x - 5y = 10 \]---
Practical Applications of Line Equations
Understanding how to derive the equation of a line has numerous real-world applications:
1. Engineering and Design
Designing structures and mechanical parts often requires precise calculations of lines and slopes.2. Business and Economics
Linear models are used for trend analysis, forecasting, and cost-profit calculations.3. Physics
Analyzing motion, velocity, and acceleration often involves linear equations.4. Navigation and Mapping
Determining routes, distances, and positions relies on line equations.---
Common Mistakes and Tips for Correctly Finding the Line Equation
To ensure accuracy, keep in mind these tips:
- Always verify the slope before plugging into formulas.
- When simplifying fractions, reduce to simplest form for clarity.
- Check your work by plugging in the original point to see if it satisfies the equation.
- Be cautious with signs; positive and negative slopes affect the line’s orientation.
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Summary: How to Find the Equation of a Line Given a Point and Slope
In summary, to find the equation of a line passing through a specific point with a given slope:
- Use the point-slope formula:
- Substitute \( (x1, y1) \) and \( m \).
- Simplify to slope-intercept form or standard form as needed.
- Verify by substituting the known point.
Applying these steps to the initial problem:
- Point: (5, 4)
- Slope: 6/5
Leads to the final equation:
\[ y = \frac{6}{5}x - 2 \]
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Conclusion
Understanding how to determine the equation of a line passing through a point with a specific slope is fundamental in algebra and geometry. It provides foundational skills for solving more complex mathematical problems and has widespread applications across various fields. By mastering the point-slope formula and practicing with different points and slopes, you can confidently derive line equations, interpret their meaning, and apply them to real-world scenarios.
Whether you are a student preparing for exams, a professional working in engineering, or simply someone interested in mathematical concepts, knowing how to find the line equation through a given point and slope is an essential skill that forms the backbone of many analytical tasks in mathematics.
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