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Magazine Chapter 8: Problem 13
Find the slope of the line determined by each pair of points. $$(-6,-1),(-2,-7)$$
Short Answer
Expert verified
The slope of the line is \
-\frac{3}{2}.
Step by step solution
01
Identify Points
Before calculating the slope, identify the given points as \((-6, -1)\) and \((-2, -7)\). Let \( (x_1, y_1) = (-6, -1) \) and \( (x_2, y_2) = (-2, -7) \).
02
Understand the Slope Formula
Recall that the slope \( m \)of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\)is given by the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \].
03
Substitute Values
Substitute the values of the points \\( (x_1, y_1) = (-6, -1) \)and \( (x_2, y_2) = (-2, -7) \)into the slope formula.\[ m = \frac{-7 - (-1)}{-2 - (-6)} \].
04
Simplify the Numerator
Calculate the difference in the numerator:\(-7 - (-1) = -7 + 1 = -6\).
05
Simplify the Denominator
Calculate the difference in the denominator:\(-2 - (-6) = -2 + 6 = 4\).
06
Calculate the Slope
Substitute the simplified numerator and denominator back into the equation:\[ m = \frac{-6}{4} \].Then simplify the fraction:\[ m = -\frac{3}{2} \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses a coordinate system to describe the precise location and properties of geometric figures. It allows for the representation of geometrical shapes, like lines and curves, using algebraic equations. This creates a bridge between algebra and geometry.
- The coordinate plane is defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
- Each point on this plane is described by an ordered pair \( (x, y) \), where \( x \) represents the horizontal distance from the origin and \( y \) represents the vertical distance.
- Understanding and plotting these points help in determining the shape, length, and orientation of lines and curves on the plane. For example, the points \((-6,-1)\) and \((-2,-7)\) describe a straight line when connected.
Slope Formula
The slope of a line is a measure of how steep that line is. It is a key concept in coordinate geometry as it describes the direction and steepness of a line. In mathematical terms, slope is often referred to as \( m \).
- The slope formula is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
- This equation determines the rate at which a line rises or falls between two points, \( (x_1, y_1) \) and \( (x_2, y_2) \).
- Positive slope means the line rises from left to right, while a negative slope means it falls.
- A zero slope indicates a horizontal line, and an undefined slope indicates a vertical line.
Simplifying Fractions
Once you've found a fraction, simplifying it is crucial to express it in its most concise form. The process of simplifying fractions involves reducing the fraction to the lowest terms in such a way that both the numerator and the denominator share no common factors other than 1. This process is often applied in coordinate geometry when calculating slopes.
- Consider the fraction \( \frac{-6}{4} \) from the slope calculation: here, both \( -6 \) and \( 4 \) can be divided by their greatest common divisor, which is 2.
- By dividing both the numerator and denominator by 2, the fraction simplifies to \( -\frac{3}{2} \).
- Always check for the greatest common divisor (GCD) when simplifying fractions to ensure accuracy.
- A simplified slope is easier to interpret and use in further calculations or graphing tasks.
