Problem 30 - Understanding Elementary Algebra with Geometry

Question

Find the slope of the line passing through the given points. Round to the nearest hundredth where necessary. \((-8.65,2.8)\) and \((12.5,-3.72)\)

Step-by-Step Solution

Answer

The slope is approximately \ -0.31 \.

Step 1 - Identify the points

The given points are \((-8.65, 2.8)\) and \((12.5, -3.72)\). Label the first point as \( (x_1, y_1) \) and the second point as \( (x_2, y_2) \). So, \( x_1 = -8.65 \), \( y_1 = 2.8 \), \( x_2 = 12.5 \), and \( y_2 = -3.72 \).

Step 2 - Write the formula for the slope

The slope \( m \) of the 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 3 - Substitute the values into the formula

Substitute the given values into the slope formula: \[ m = \frac{-3.72 - 2.8}{12.5 - (-8.65)} \].

Step 4 - Simplify the numerator

Calculate the difference in the \( y \)-coordinates: \( -3.72 - 2.8 = -6.52 \). So, the new formula becomes: \[ m = \frac{-6.52}{12.5 - (-8.65)} \].

Step 5 - Simplify the denominator

Calculate the difference in the \( x \)-coordinates: \( 12.5 - (-8.65) = 12.5 + 8.65 = 21.15 \). So, the new formula is: \[ m = \frac{-6.52}{21.15} \].

Step 6 - Calculate the slope

Divide \( -6.52 \) by \( 21.15 \): \[ m \approx -0.31 \], rounding to the nearest hundredth.

Key Concepts

slope formula

In coordinate geometry, the **slope** of a line measures its steepness or incline. To find the slope between two points, we use the **slope formula**: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Here, \( x_1 \) and \( y_1 \) are the coordinates of the first point, and \( x_2 \) and \( y_2 \) are the coordinates of the second point. By subtracting the first point's coordinates from the second point's, we get the rise over the run. This difference tells us how much the line goes up or down between the two points and how far it goes horizontally. In our example:

• \( x_1 = -8.65 \)
• \( y_1 = 2.8 \)
• \( x_2 = 12.5 \)
• \( y_2 = -3.72 \)

We plug these values into the formula to find the slope between these two points.

coordinate geometry

Coordinate geometry, or analytic geometry, is a branch of mathematics where geometric problems are solved using algebraic formulas involving coordinates. Every point on the plane is represented as a pair \((x, y)\). To solve problems in this field, you need to:

• Understand how to plot points
• Use coordinates to derive distances and slopes
• Find equations of geometric figures like lines and circles

Knowing how to calculate the slope is essential for understanding the properties of lines. It also helps in finding parallel and perpendicular lines, which are key concepts in coordinate geometry. The slope tells us how tilted a line is on the Cartesian plane, which helps with interpreting graphs and solving application problems.

linear equations

A **linear equation** represents a straight line when plotted on a coordinate plane. It typically takes the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. From the slope calculation, finding the slope \( m \) is crucial because it tells us how fast \( y \) changes with respect to \( x \). To form a linear equation:

• Determine the slope using the slope formula
• Identify a point on the line (usually given)
• Use the slope and the point to solve for the y-intercept \( b \)

For example, if you have a line passing through points \((-8.65, 2.8)\) and \((12.5, -3.72)\), finding the slope using the given formula gives us insight into the line's equation. These equations are fundamental for solving many types of mathematical and real-world problems where relationships between variables need to be understood and quantified.