Problem 13 - Elementary Algebra

Question

Find the slope of the line determined by each pair of points. $$(-6,-1),(-2,-7)$$

Step-by-Step Solution

Answer

The slope of the line is \ -\frac{3}{2}.

Step 1: Identify Points

Before calculating the slope, identify the given points as \((-6, -1)\) and \((-2, -7)\). Let \( (x_1, y_1) = (-6, -1) \) and \( (x_2, y_2) = (-2, -7) \).

Step 2: 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:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \].

Step 3: 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.\[ m = \frac{-7 - (-1)}{-2 - (-6)} \].

Step 4: Simplify the Numerator

Calculate the difference in the numerator:\(-7 - (-1) = -7 + 1 = -6\).

Step 5: Simplify the Denominator

Calculate the difference in the denominator:\(-2 - (-6) = -2 + 6 = 4\).

Step 6: Calculate the Slope

Substitute the simplified numerator and denominator back into the equation:\[ m = \frac{-6}{4} \].Then simplify the fraction:\[ m = -\frac{3}{2} \].

Key Concepts

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.

This connection between graphical and algebraic representations is central in solving problems related to distance, midpoints, and particularly, the slope of a line.

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.

By using the given points \((-6,-1)\) and \((-2,-7)\), the slope formula becomes a practical way to determine that the line between these points has a specific negative slope, reflecting the line’s decrease as you move from left to right across the graph.

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.

This simplification not only makes calculations more straightforward but also helps maintain clarity and understanding in solving geometric problems.