Problem 27 - Understanding Elementary Algebra with Geometry

Question

Find the slope of the line passing through the given points. Round to the nearest hundredth where necessary. \((0.8,2.65)\) and \((1.3,4.72)\)

Step-by-Step Solution

Answer

The slope is 4.14.

Step 1 - Identify the coordinates

Identify the given points. In this problem, the points are \((0.8, 2.65)\) and \((1.3, 4.72)\). Name the points as follows: \(x_1 = 0.8, y_1 = 2.65\) and \(x_2 = 1.3, y_2 = 4.72\).

Step 2 - Use the slope formula

The formula to find the slope \(m\) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substitute the coordinates into the formula.

Step 3 - Substitute the coordinates into the formula

Substitute \(x_1 = 0.8\), \(y_1 = 2.65\), \(x_2 = 1.3\), and \(y_2 = 4.72\) into the slope formula: \[ m = \frac{4.72 - 2.65}{1.3 - 0.8}\]

Step 4 - Perform the subtraction

First, perform the subtraction in the numerator: \[ 4.72 - 2.65 = 2.07 \] Then, perform the subtraction in the denominator: \[ 1.3 - 0.8 = 0.5 \]

Step 5 - Divide the results

Divide the result of the numerator by the result of the denominator to find the slope: \[ m = \frac{2.07}{0.5} = 4.14 \]

Key Concepts

Elementary Algebra

Elementary algebra is the foundation for understanding more complex mathematical concepts. It's often what students first learn and is essential for solving basic equations.

In this context, we're dealing with simple algebraic operations like subtraction and division.

For example, given the points \( (0.8, 2.65) \) and \( (1.3, 4.72) \), we subtract the y-coordinates and x-coordinates separately.

This subtraction must be done accurately to find the slope correctly.

Understanding elementary algebra helps students become proficient at these basic operations, which are crucial building blocks for more advances topics.

Linear Equations

Linear equations involve relationships that can be represented by a straight line on a graph.

The slope of this line is a key characteristic, which indicates how steep the line is.

The formula used to find the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].

By substituting our given points into the formula, we can determine the rate of change or 'steepness' of the line.

This allows us to understand and predict how one variable changes with respect to another, which is a fundamental aspect of linear equations.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to represent geometric figures like points and lines using algebraic equations.

Using a coordinate plane, we can plot points \( (0.8, 2.65) \) and \( (1.3, 4.72) \).

The slope between these points tells us the direction and steepness of the line connecting them.

In coordinate geometry, the slope is a crucial concept because it defines the orientation of a line in the plane.

Understanding how to calculate and interpret the slope can greatly enhance your comprehension of spatial relationships.

Calculation Methods

Various calculation methods are used to find the slope of a line, and one of the most straightforward is through manual computation.

Here's a step-by-step method:
• Identify the coordinates: Let's call them \( x_1 = 0.8, y_1 = 2.65 \) and \( x_2 = 1.3, y_2 = 4.72 \).

• Use the slope formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]; substitute the coordinates into this formula.

• First, subtract the y-coordinates: \[ 4.72 - 2.65 = 2.07 \]

• Next, subtract the x-coordinates: \[ 1.3 - 0.8 = 0.5 \]

• Finally, divide the two results: \[ m = \frac{2.07}{0.5} = 4.14 \]

Following these steps carefully ensures accuracy in your calculations and helps you grasp the procedural aspect of slope determination.