Problem 27 - Algebra 2

Question

Find the slope of the line that passes through each pair of points. $$ (5,4),(-3,8) $$

Step-by-Step Solution

Answer

The slope of the line is \\(-\frac{1}{2}\\).

Step 1: Identify the Point Coordinates

First, identify the coordinates of the two points. The first point is \(5, 4\) and the second point is \((-3, 8)\).

Step 2: Recall the Slope Formula

The formula for the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \((y_2 - y_1) / (x_2 - x_1)\).

Step 3: Substitute the Coordinates

Substitute the coordinates into the slope formula: \(x_1 = 5\), \(y_1 = 4\), \(x_2 = -3\), and \(y_2 = 8\). Then calculate: \[ m = \frac{8 - 4}{-3 - 5} = \frac{4}{-8} \].

Step 4: Simplify the Result

Simplify the fraction \(\frac{4}{-8}\) to its simplest form. It reduces to \(\frac{-1}{2}\). Hence, the slope of the line is \(-\frac{1}{2}\).

Key Concepts

Coordinate Geometry

Coordinate Geometry is a branch of mathematics that helps us understand the layout and relationships between different points on a plane. Basically, it deals with the study of geometry using a coordinate system. The standard coordinate system, also known as the Cartesian coordinate system, consists of two axes: the x-axis (horizontal) and the y-axis (vertical). Any point in this system is represented by a pair of numbers, which are its x and y coordinates. For example, the points (5, 4) and (-3, 8) are such coordinates.

The pair of numbers tells us where the point is located in relation to these axes. The first number is always for the x-coordinate and the second one for the y-coordinate. When plotting these points, you can see them positioned uniquely, which helps in identifying properties like distance and slope of a line passing through these points.

Coordinate geometry allows us to draw mathematical connections between geometric figures and algebraic equations. Each line, curve, or shape can be described by a mathematical equation, thus marrying the world of geometry with algebra.

Slope Formula

The slope of a line is a measure of its steepness and direction. It is a crucial concept in coordinate geometry as it tells us how inclined a line is. The formula to determine the slope (m) between two points, e.g., \( (x_1, y_1) \) and \( (x_2, y_2) \), is:

• m = (\( y_2 - y_1 \))/(\( x_2 - x_1 \))

This formula subtracts the y-coordinates of the two points and divides by the subtraction of the x-coordinates.

In our exercise, we've got two points: (5,4) and (-3,8). Applying the slope formula means plugging these coordinates into the equation:

• \( y_2 = 8 \),\( y_1 = 4 \)
• \( x_2 = -3 \),\( x_1 = 5 \)

This calculates as follows:

• m = (8 - 4)/(-3 - 5)
• m = 4/-8

This results in a slope value that describes the shift in vertical placement of one point divided by the change in horizontal placement, giving insight into how the line inclines or declines.

Simplifying Fractions

Simplifying fractions is an arithmetic method to reduce expressions to their simplest form, making them easier to handle. A fraction is simplified when you can no longer divide the numerator and the denominator by any number other than 1 (or -1 for negative fractions).

In this exercise, once we calculated the slope as a fraction, it came out as \( \frac{4}{-8} \). Both 4 and 8 can be divided by 4, which is their greatest common divisor (GCD), to simplify the fraction:

• Divide the numerator by 4 (\( 4 \div 4 = 1 \))
• Divide the denominator by 4 (\( -8 \div 4 = -2 \))

Thus, \( \frac{4}{-8} \) simplifies to \( \frac{-1}{2} \). Therefore, the slope is simplified to \(-\frac{1}{2}\).

Simplifying fractions not only makes them cleaner but often allows for easier interpretation and comparison when applying them further in larger mathematical contexts.