Find The Slope Of The Given Line, If It Is Defined. The Line Through The Origin And (-9,12) Select The
Understanding how to find the slope of a line is fundamental in coordinate geometry. When given two points, such as the line passing through the origin (0, 0) and another point (-9, 12), calculating the slope provides insight into the line’s steepness and direction. This article offers a comprehensive guide to determining the slope of such a line, explaining the concept step-by-step, including relevant formulas, examples, and applications, all structured for clarity and SEO optimization.
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What Is the Slope of a Line?
The slope of a line measures its inclination or steepness. It indicates how much the y-coordinate (vertical change) of the line changes concerning the x-coordinate (horizontal change). The slope is typically denoted by the letter m.
Key points about slope:
- The slope is a ratio of the "rise" over the "run."
- It can be positive, negative, zero, or undefined.
- A positive slope means the line ascends from left to right.
- A negative slope indicates the line descends from left to right.
- A zero slope corresponds to a horizontal line.
- An undefined slope corresponds to a vertical line.
Understanding these concepts is essential before calculating the slope of the line passing through specific points.
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How to Find the Slope of a Line Given Two Points
The general method to find the slope when two points are known involves using the slope formula:
\[ m = \frac{y2 - y1}{x2 - x1} \]
where:
- \((x1, y1)\) are the coordinates of the first point.
- \((x2, y2)\) are the coordinates of the second point.
This formula calculates the ratio of the change in y-values to the change in x-values between the two points.
Steps to find the slope:
- Identify the coordinates of the two points.
- Subtract the y-coordinate of the first point from the y-coordinate of the second point.
- Subtract the x-coordinate of the first point from the x-coordinate of the second point.
- Divide the difference in y by the difference in x.
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Calculating the Slope for the Line Through the Origin and (-9, 12)
Given the problem:
- First point: Origin (0, 0)
- Second point: (-9, 12)
We will apply the slope formula:
\[ m = \frac{12 - 0}{-9 - 0} \]
Calculating numerator and denominator:
\[ m = \frac{12}{-9} \]
Simplify the fraction:
\[ m = -\frac{12}{9} = -\frac{4}{3} \]
Thus, the slope of the line passing through the origin and (-9, 12) is \(-\frac{4}{3}\).
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Interpreting the Slope \(-\frac{4}{3}\)
The slope value provides valuable information about the line’s behavior:
- Negative slope: The line descends from left to right.
- Magnitude \(\frac{4}{3}\): For every 3 units it moves horizontally to the right (positive x-direction), it moves 4 units downward (negative y-direction).
Implications:
- The line is inclined downward as we move along the positive x-axis.
- The steepness is moderate; it’s neither flat nor almost vertical.
- The slope indicates a consistent rate of change between x and y coordinates.
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Equation of the Line Through the Origin and (-9, 12)
Once the slope is known, the equation of the line can be written using the point-slope form:
\[ y - y1 = m(x - x1) \]
Since the line passes through the origin (0, 0), the point-slope form simplifies to:
\[ y = m x \]
Substituting the slope:
\[ y = -\frac{4}{3} x \]
This is the equation of the line passing through (0, 0) and (-9, 12).
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Verification: Confirming the Line Passes Through (-9, 12)
To verify, substitute \(x = -9\) into the equation:
\[ y = -\frac{4}{3} \times (-9) \]
\[ y = -\frac{4}{3} \times -9 = \frac{4}{3} \times 9 \]
\[ y = 4 \times 3 = 12 \]
Since this matches the y-coordinate of the second point, the calculation is correct, confirming the line passes through both points.
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Applications of Finding the Slope of a Line
Understanding how to find the slope has numerous practical applications in various fields:
- Physics: Calculating velocity (rate of change of position over time).
- Economics: Analyzing cost functions and marginal analysis.
- Engineering: Designing inclined planes or ramps.
- Mathematics: Solving systems of equations and analyzing linear relationships.
- Data Science: Determining correlations between variables.
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Additional Tips for Calculating Slope
- Always check whether the denominator (\(x2 - x1\)) is zero to avoid division by zero, which indicates a vertical line with an undefined slope.
- Simplify fractions to understand the steepness better.
- Remember that the slope is unaffected by the line’s position; only the direction and steepness matter.
Common Mistakes to Avoid
- Swapping the points, which can lead to a different sign for the slope.
- Forgetting to reduce fractions for clarity.
- Confusing the points’ coordinates, especially when points are given in different formats.
- Ignoring vertical or horizontal lines, which have special slope considerations.
Conclusion
Determining the slope of a line passing through two points, especially when one point is the origin, is a straightforward process involving the slope formula \(\frac{y2 - y1}{x2 - x1}\). For the line passing through (0, 0) and (-9, 12), the slope is \(-\frac{4}{3}\), indicating a downward-sloping line with a moderate incline. Recognizing this value allows for the derivation of the line’s equation and understanding of its geometric behavior. Mastery of slope calculation is essential in algebra, calculus, and real-world problem-solving, providing foundational knowledge for more complex mathematical concepts.
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Meta Description:
Learn how to find the slope of a line passing through the origin and point (-9, 12). Step-by-step explanation, formula application, and practical insights for understanding line slopes in coordinate geometry.
