Problem 75 - Intermediate Algebra

Question

Find the slope of each line. The line that passes through \((0,-8)\) and \((-5,0)\)

Step-by-Step Solution

Answer

The slope of the line is \(-\frac{8}{5}\).

Step 1: Identify Coordinates

First, identify the given coordinates of the two points. In this exercise, the points are \((x_1, y_1) = (0, -8)\) and \((x_2, y_2) = (-5, 0)\).

Step 2: Use the Slope Formula

The formula for the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]

Step 3: Substitute the Coordinates into the Formula

Substitute the coordinates \((0,-8)\) and \((-5,0)\) into the slope formula: \[m = \frac{0 - (-8)}{-5 - 0} = \frac{8}{-5}\]

Step 4: Simplify the Fraction

Simplify the fraction obtained:\[m = \frac{8}{-5} = -\frac{8}{5}\].The negative sign indicates the slope is negative.

Key Concepts

Understanding Coordinates

Coordinates are a fundamental concept in geometry and mathematics, used to pinpoint the position of a point on a plane. They are often written as ordered pairs \((x, y)\), where "x" represents a position along the horizontal axis and "y" represents a position along the vertical axis.

• The first number in the ordered pair—known as the x-coordinate—tells us how far to move horizontally from the origin.
• The second number, or y-coordinate, tells us how far to move vertically.

In our example with the points \((0, -8)\) and \((-5, 0)\), the coordinates tell us exactly where these two points are on the coordinate plane. The first point is directly on the y-axis, and the second is on the x-axis.
Understanding how to read and locate these points helps immensely when calculating distances, slopes, and other geometric values.

Applying the Slope Formula

The slope of a line is a measure of its steepness and direction. To find it, you use the slope formula, which compares the "rise" (change in y) to the "run" (change in x) of the line.
The formula is expressed as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two points on the line.

• The numerator \((y_2 - y_1)\) represents how much the line goes up or down.
• The denominator \((x_2 - x_1)\) represents how much the line goes left or right.

Inserting the coordinates \((0, -8)\) and \((-5, 0)\) into the formula gives us: \[ m = \frac{0 - (-8)}{-5 - 0} \] This simplifies to \( m = \frac{8}{-5} \), showing that the line is falling as you move from left to right.

Simplifying Fractions in Slope

Simplifying fractions is crucial for expressing numbers in their simplest form. When you compute the slope, you often find yourself with a fraction that needs simplifying.
Let's revisit our slope calculation example: \[ m = \frac{8}{-5} \] This fraction is already simplest in terms of numerator and denominator, but the negative sign can be confusing.

• A negative sign in a fraction can be placed with either the numerator, the denominator, or in front of the entire fraction, so \(\frac{8}{-5} = -\frac{8}{5}\).
• This means that for every 8 units the line rises, it moves 5 units to the left, indicating the line's downward slope.

Keeping the slope in its simplest form helps in interpreting the mathematical relationship it represents, making it easier to graph and understand.