Problem 20 - Elementary Algebra

Question

Find the slope of the line determined by each pair of points. $$(a, 0),(0, b)$$

Step-by-Step Solution

Answer

The slope is \(-\frac{b}{a}\).

Step 1: Recall the Formula for Slope

The slope (m) of a line passing through two 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 2: Identify the Points (a,0) and (0,b)

In this problem, we are given the points: \((x_1, y_1) = (a, 0)\) and \((x_2, y_2) = (0, b)\).

Step 3: Substitute the Points into the Slope Formula

Substitute \((x_1, y_1) = (a, 0)\) and \((x_2, y_2) = (0, b)\) into the slope formula:\[m = \frac{b - 0}{0 - a} = \frac{b}{-a}\]

Step 4: Simplify the Expression

Simplify the expression for the slope:\[m = -\frac{b}{a}\]

Key Concepts

Two Points

When we talk about two points in coordinate geometry, we are referring to specific locations on a graph. Each point is defined by a pair of coordinates: an x-coordinate, which tells us how far to move left or right, and a y-coordinate, which tells us how far to move up or down. For example, in the exercise, we have the points

• \((a,0)\) where \(a\) is the x-coordinate and 0 is the y-coordinate
• \((0,b)\) where 0 is the x-coordinate and \(b\) is the y-coordinate

These coordinates help us locate the exact position of each point on a standard two-dimensional plane. Knowing the positions of these points is crucial for determining the slope of the line that passes through them.

Slope Formula

The slope of a line is a measure of its steepness, and can be calculated using two points on the line. The formula for the slope \(m\) is given by: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here:

• \(y_2\) and \(y_1\) are the y-coordinates of the two points
• \(x_2\) and \(x_1\) are the x-coordinates of the two points

Plugging these values into the formula allows us to calculate how much y changes for each unit change in x. In simpler terms, the slope tells us whether the line rises, falls, or remains constant. A positive slope indicates the line rises, a negative slope means it falls, and a slope of zero means the line is flat.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves using algebra to study and represent geometric concepts. It places geometric shapes on a graph using a coordinate system.In this system, each point is represented by an ordered pair \((x, y)\) that corresponds to its location on a plane. This makes it easy to understand relationships between different geometric figures. By applying coordinate geometry, we can compute distances, slopes, and more. It forms the basis for creating linear equations and understanding how they dictate the graph's direction and shape. It is an invaluable tool for solving problems involving lines, circles, and other shapes by translating them into algebraic equations.

Linear Equations

Linear equations represent straight lines on a graph. They can be formulated once we know the slope and a point the line passes through. The most common form used is the slope-intercept form: \[y = mx + c\]Here:

• \(m\) is the slope of the line, indicating its steepness
• \(c\) is the y-intercept, which is where the line crosses the y-axis

Upon finding the slope from two points, we can plug it into this formula along with one point to find the y-intercept. This equation can then be used to predict the y-value for any x-value, showcasing the line's behavior across the graph.