Problem 28 - Understanding Elementary Algebra with Geometry

Question

Find the slope of the line passing through the given points. Round to the nearest hundredth where necessary. \((12.63,10.44)\) and \((9.48,7.96)\)

Step-by-Step Solution

Answer

The slope is approximately 0.79.

Step 1: Identify the points

Let's label the given points as \(x_1, y_1\) and \(x_2, y_2\). Here, the points are \( (12.63, 10.44) \) and \( (9.48, 7.96) \). So, \( x_1 = 12.63 \), \( y_1 = 10.44 \), \( x_2 = 9.48 \), and \( y_2 = 7.96 \).

Step 2: Write 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 \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Step 3: Plug in the values into the formula

Substitute \( x_1 = 12.63 \), \( y_1 = 10.44 \), \( x_2 = 9.48 \), and \( y_2 = 7.96 \) into the slope formula: \[ m = \frac{7.96 - 10.44}{9.48 - 12.63} \]

Step 4: Simplify the numerator and denominator

Calculate the differences in the numerator and denominator separately: \[ y_2 - y_1 = 7.96 - 10.44 = -2.48 \] \[ x_2 - x_1 = 9.48 - 12.63 = -3.15 \]

Step 5: Divide the results

Now, divide the differences to find the slope: \[ m = \frac{-2.48}{-3.15} \] \[ m \approx 0.79 \]

Step 6: Round to the nearest hundredth

The slope rounded to the nearest hundredth is: \[ m \approx 0.79 \]

Key Concepts

coordinate geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometric figures using coordinates on a plane. It allows us to use algebraic equations and formulas to solve geometric problems. By using a coordinate plane, each point is defined by an \(x, y\) pair, where \(x\) represents the horizontal position and \(y\) the vertical position.

In this exercise, we are given two points, \( (12.63, 10.44) \) and \( (9.48, 7.96) \), and are asked to find the slope of the line passing through them. Each point represents a location in coordinate geometry. Identifying these coordinates is essential for applying algebraic methods.

slope formula

The slope of a line measures how steep the line is. It is calculated using the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

This formula tells us the change in \(y\)-coordinates divided by the change in \(x\)-coordinates between two points on the line.

Let's break down the components:

• \