Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(23-28\), find the slope of the line containing the given two points. (3,0) \text { and }(0,5)

Step-by-Step Solution

Answer

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

Step 1: Understand the Slope Formula

The slope of a line is a measure of its steepness. To find the slope when given two points, use the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

Step 2: Identify the Points

From the problem, we have two points: \((3, 0)\) and \((0, 5)\). Here, \(x_1 = 3\), \(y_1 = 0\), \(x_2 = 0\), and \(y_2 = 5\). Now, we will plug these values into the slope formula.

Step 3: Substitute Points into the Slope Formula

Substitute the values into the slope formula:\[ m = \frac{5 - 0}{0 - 3} \]

Step 4: Simplify the Expression

Now, perform the arithmetic to simplify the expression:\[ m = \frac{5}{-3} \]Thus, the slope of the line is \(-\frac{5}{3}\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that deals with geometric objects using coordinate systems. One of its essential components is understanding the slope of a line. The slope tells us how fast a line rises or falls as you move along it.
In coordinate geometry, you often work with points, which are usually represented as ordered pairs like \((x, y)\). These coordinates allow you to place points on a plane and investigate their properties and relationships using algebraic equations.
When finding the slope between two points, imagine a right triangle being formed with the line segment as the hypotenuse. The change in the \(y\)-coordinates over the change in the \(x\)-coordinates between two points will give you the slope, which is essential for understanding the behavior of linear equations.

Linear Equations

Linear equations are equations of the first degree, meaning they involve variables raised to the power of one. They are represented graphically as straight lines when plotted on a coordinate plane.
When solving problems involving linear equations, figuring out the slope is critical. The slope-intercept form of a linear equation, \(y = mx + b\), showcases this: \(m\) represents the slope, and \(b\) represents the y-intercept.

• The slope \(m\) is a measure of how the line rises or falls.
• The y-intercept \(b\) is the point where the line crosses the y-axis.

Solving for these elements helps to quickly graph lines and understand the relationship between variables. In coordinate geometry exercises, calculating the slope helps check if different lines are parallel (having the same slope) or perpendicular (having slopes that are negative reciprocals).

Mathematics Problem Solving

Mathematics problem solving often requires a systematic approach to breaking down complex issues into simpler parts. In the case of finding the slope of a line given two points, it involves several key steps:

• Understand the problem: Identify what is being asked, in this case, finding the slope.
• Analyze the data: Recognize the given points and their coordinates.
• Apply formulas: Use the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) to calculate the slope.
• Calculate and conclude: Simplify the expression to find the slope, ensuring all arithmetic is correct.

Developing these skills not only helps in solving coordinates geometry problems effectively but also improves logical thinking and the ability to tackle mathematical challenges in a structured way.