Problem 25 - Algebra for College Students

Question

Find the slope of the line that contains each of the following pairs of points. \((3,-5),(0,0)\)

Step-by-Step Solution

Answer

The slope is \( -\frac{5}{3} \).

Step 1 - Understand the Formula for Slope

The slope of a line is determined by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here \( (x_1, y_1) \) and \( (x_2, y_2) \) are coordinates of two points on the line.

Step 2 - Identify the Coordinates

Identify the coordinates of the given points. Here, \( (3, -5) \) and \( (0, 0) \). Assign \( (3, -5) \) to \( (x_1, y_1) \) and \( (0, 0) \) to \( (x_2, y_2) \).

Step 3 - Substitute the Values into the Formula

Substitute \( x_1 = 3 \), \( y_1 = -5 \), \( x_2 = 0 \), and \( y_2 = 0 \) into the slope formula: \( m = \frac{0 - (-5)}{0 - 3} \).

Step 4 - Simplify the Expression

Simplify the expression \( m = \frac{0 - (-5)}{0 - 3} = \frac{5}{-3} = -\frac{5}{3} \).

Step 5 - State the Slope

The slope of the line passing through the points \( (3, -5) \) and \( (0, 0) \) is \( -\frac{5}{3} \).

Key Concepts

Coordinates

To understand the slope of a line, it is essential to know about coordinates. Coordinates are ordered pairs \((x, y)\) that represent points on a coordinate plane. In the given exercise, we have two points: \((3, -5)\) and \((0, 0)\). The first value in each pair is the x-coordinate, which indicates the horizontal position. The second value is the y-coordinate, indicating the vertical position.
By using coordinates, we can precisely locate points and determine the relationship between them. This relationship can then be analyzed using various mathematical formulas and methods, like finding the slope.

Slope Formula

The slope of a line is a measure of its steepness and direction. The formula to calculate the slope (\(m\)) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\(\text{slope} = m = \frac{y_2 - y_1}{x_2 - x_1} \)
This formula calculates the ratio of the vertical change (rise) to the horizontal change (run) between the two points.

• The vertical change (\(y_2 - y_1\)) measures how much the y-coordinate changes between the points.
• The horizontal change (\(x_2 - x_1\)) measures how much the x-coordinate changes.

Using the given points \((3, -5)\) and \((0, 0)\), we substitute:
\(\text{Slope} = m = \frac{0 - (-5)}{0 - 3} = \frac{5}{-3} = -\frac{5}{3} \)

Algebraic Simplification

After substituting the values into the slope formula, it's crucial to simplify the resulting expression. Algebraic simplification involves performing basic operations to reduce the expression to its simplest form.
For the given points, the formula becomes: \( m = \frac{0 - (-5)}{0 - 3} = \frac{5}{-3} \).
Here, we take the difference in y-coordinates (rise) and x-coordinates (run), then perform the division. Simplifying \(\frac{5}{-3}\) gives us the final slope:
\(\text{slope} = -\frac{5}{3} \). This negative result indicates that the line slopes downward from left to right. Simplification ensures that the answer is clear and concise, making it easier to interpret and use in further calculations or graphing.