A Line Passes Through The Points ( 1,2) And (3,5)
Understanding how to find the equation of a line passing through two points is a fundamental concept in coordinate geometry. When given the points (1, 2) and (3, 5), mathematicians and students alike can determine the precise mathematical description of the line that connects them. This process involves calculating the slope of the line and then using point-slope or slope-intercept form to express its equation. In this article, we will explore the step-by-step method to derive the equation of the line passing through these points, discuss key concepts like slope and intercepts, and provide practical examples to solidify understanding.
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Understanding the Basics: Points, Coordinates, and Lines
Before delving into calculations, it's crucial to grasp some foundational concepts:
Coordinate Points
- Definition: Points in a plane are represented by ordered pairs (x, y), where 'x' is the horizontal coordinate and 'y' is the vertical coordinate.
- Example: The points (1, 2) and (3, 5) are two locations on the Cartesian plane, with their respective x and y values.
Line in Coordinate Geometry
- Equation of a line: It describes all the points that lie along a straight path.
- Standard forms include:
- Slope-intercept form: y = mx + b
- Point-slope form: y - y₁ = m(x - x₁)
- General form: Ax + By + C = 0
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Calculating the Slope of the Line Passing Through (1,2) and (3,5)
The first step in finding the equation of the line is computing its slope. The slope (m) indicates how steep the line is and the direction it moves.
Definition of Slope
- The slope between two points (x₁, y₁) and (x₂, y₂) is calculated as:
- It represents the rate of change of y with respect to x.
Calculating the Slope for Points (1, 2) and (3, 5)
- Assign:
- (x₁, y₁) = (1, 2)
- (x₂, y₂) = (3, 5)
- Plug into the formula:
- Result: The slope of the line passing through the points (1, 2) and (3, 5) is m = 3/2.
Deriving the Equation of the Line
Once the slope is known, the next step is to find the actual equation of the line in a usable form.
Using the Point-Slope Form
- The point-slope form is particularly convenient when you know a point on the line and the slope:
- Choose one of the given points, say (1, 2), and plug in the slope:
Simplifying to Slope-Intercept Form
- Expand the equation:
- Add 2 to both sides (or convert 2 to a fraction):
- Convert 2 to a fraction with denominator 2:
- So:
Final Equation:
\[
\boxed{ y = \frac{3}{2}x + \frac{1}{2} }
\]
This is the equation of the line passing through (1, 2) and (3, 5).
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Verifying the Equation with the Second Point
It's important to verify that the derived equation passes through both points.
Check with (3, 5):
- Substitute x = 3:
- Since y = 5, the point (3, 5) satisfies the equation.
Check with (1, 2):
- Substitute x = 1:
- y = 2 confirms the point (1, 2) also lies on the line.
Graphing the Line: Visual Representation
Visualizing the line helps in understanding its position and slope. Here’s how you can do it:
Steps to Plot the Line
- Plot the points (1, 2) and (3, 5) on graph paper or graphing software.
- Draw a straight line through these points.
- Note the slope: for every 2 units moved horizontally, the line rises 3 units vertically.
- Extend the line in both directions to visualize its full span.
Key Features to Note
- Slope: 3/2 signifies the line rises 3 units for every 2 units moved horizontally.
- Y-intercept: The point where the line crosses the y-axis is at (0, 0.5), which can be found by setting x=0:
\[
y = \frac{3}{2} \times 0 + \frac{1}{2} = \frac{1}{2}
\]
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Applications and Practical Examples
Understanding this fundamental concept has numerous practical applications:
Real-World Applications of Line Equations
- Physics: Calculating the trajectory of objects.
- Economics: Modeling cost versus production quantities.
- Engineering: Designing structures with specific slope requirements.
- Data Analysis: Fitting a line to data points for trend analysis.
Example Problem: Finding the Equation of a Line through Two Data Points
Suppose a company tracks sales over two months:
- Month 1: 100 units sold.
- Month 2: 250 units sold.
Find the line equation modeling sales over these months, assuming month numbers as x (Month 1 = x=1, Month 2 = x=2), and units sold as y.
Solution:
- Points: (1, 100) and (2, 250)
- Calculate slope:
\[
m = \frac{250 - 100}{2 - 1} = \frac{150}{1} = 150
\]
- Use point-slope form with (1, 100):
\[
y - 100 = 150(x - 1)
\]
- Expand:
\[
y - 100 = 150x - 150
\]
- Simplify to slope-intercept form:
\[
y = 150x - 150 + 100 = 150x - 50
\]
Interpretation: The sales increase by 150 units each month, starting from a baseline that can be interpreted as -50 (which may require context adjustment).
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Conclusion
Finding the equation of a line passing through two points, such as (1, 2) and (3, 5), is a vital skill in coordinate geometry. The process involves calculating the slope, verifying it, and then expressing the line in an appropriate form—most commonly slope-intercept. This method provides a clear mathematical representation of the line and supports various applications across science, engineering, and economics. By mastering these concepts, students and professionals can analyze and interpret linear relationships effectively, laying a solid foundation for more advanced mathematical topics.
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Additional Tips for Learning and Teaching Line Equations
- Always verify your line equation with both points.
- Practice with different pairs of points to strengthen understanding.
- Use graphing tools to visualize the line for better intuition.
- Remember that the slope can be positive, negative, zero, or undefined, each indicating different line orientations.
