Problem 88 - Algebra A Combined Function

Question

Find the slope of the line through the given points. $$ (2.1,6.7) \text { and }(-8.3,9.3) $$

Step-by-Step Solution

Answer

The slope is -0.25.

Step 1: Identify the Slope Formula

The slope of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Where \(m\) is the slope.

Step 2: Substitute the Given Points into the Formula

Given points are \((2.1, 6.7)\) and \((-8.3, 9.3)\). Assign these as follows: \(x_1 = 2.1\), \(y_1 = 6.7\)and \(x_2 = -8.3\), \(y_2 = 9.3\).Substitute these values into the formula: \[ m = \frac{9.3 - 6.7}{-8.3 - 2.1} \]

Step 3: Calculate the Differences in Coordinates

Subtract the y-coordinates: \(9.3 - 6.7 = 2.6\).Subtract the x-coordinates: \(-8.3 - 2.1 = -10.4\).

Step 4: Compute the Slope

Divide the difference in y-coordinates by the difference in x-coordinates:\[ m = \frac{2.6}{-10.4} = -0.25 \]Thus, the slope of the line is \(-0.25\).

Key Concepts

Slope Formula

The slope of a line is a quite important concept in mathematics. It tells us how steep a line is or the direction it tilts. When we want to determine the slope using two points, we rely on the slope formula. This formula is a simple way to calculate the rate at which the y-values change compared to the x-values for any two points on a line. The slope formula is expressed as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Where:

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

To put it simply, this formula finds the difference in the y-values of the points and divides it by the difference in the x-values. A positive result means the line is rising, while a negative result means it's falling.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves studying geometry using a coordinate system. By placing figures on a plane with a grid, we use coordinates to describe points and then analyze relationships such as slopes or distances between points. The points \( (2.1, 6.7) \) and \( (-8.3, 9.3) \)in a coordinate plane make it possible to use algebra to solve geometric problems. Understanding this concept is crucial for solving various problems, like calculating the slope using coordinates. Real-world applications abound, from map reading to computer graphics, which use coordinate geometry principles to create visuals and solve problems related to space and motion.

Calculation Steps

Knowing how to perform the calculations for finding the slope of a line can make life easier. Here's an easy guide on how to determine the slope:First, outline your points. You begin with the points \( (2.1, 6.7) \) and \( (-8.3, 9.3) \). Assign these numerical values to variables:

• \( x_1 = 2.1 \), \( y_1 = 6.7 \)
• \( x_2 = -8.3 \), \( y_2 = 9.3 \)

Now, use the slope formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Plug the numbers into this formula:\[ m = \frac{9.3 - 6.7}{-8.3 - 2.1} \]Next, calculate the differences in the coordinates:

• Subtract the y-values: \( 9.3 - 6.7 = 2.6 \)
• Subtract the x-values: \( -8.3 - 2.1 = -10.4 \)

Finally, divide these differences:\[ m = \frac{2.6}{-10.4} = -0.25 \]This result means the line tilts downwards, at a slope of \(-0.25\). Understanding these steps is key to mastering slope calculations.