Problem 26 - Beginning and Intermediate Algebra

Question

Use the slope formula to find the slope of the line containing each pair of points. $$(-2,-3) \text { and }(3,-1)$$

Step-by-Step Solution

Answer

The slope of the line containing the pair of points \((-2, -3)\) and \((3, -1)\) is found using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Plugging in the given coordinates, we get \(m = \frac{-1 - (-3)}{3 - (-2)} = \frac{2}{5}\). Therefore, the slope is \(\frac{2}{5}\).

Step 1: Identify the coordinates

The given points are \((-2, -3)\) and \((3, -1)\). We can identify the coordinates as: \(x_1 = -2, y_1 = -3, x_2 = 3, y_2 = -1\)

Step 2: Plug the values into the slope formula

Now, we will plug these coordinates values into the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - (-3)}{3 - (-2)}\)

Step 3: Solve for the slope

Perform the calculations within the parentheses, then simplify the fraction: \(m = \frac{-1 + 3}{3 + 2} = \frac{2}{5}\) The slope of the line containing the pair of points \((-2, -3)\) and \((3, -1)\) is \(\frac{2}{5}\).

Key Concepts

Coordinates

Coordinates are the specific points on a graph that help us locate positions on a two-dimensional plane. Each coordinate pair consists of an 'x' value and a 'y' value, often referred to as

• x-coordinate: Placed on the horizontal axis;
• y-coordinate: Placed on the vertical axis.

These are typically written in the format \(x, y\), with the 'x' part always coming first. For example, in the point \((-2, -3)\), the 'x' coordinate is -2, and the 'y' coordinate is -3.

To solve problems like finding a line’s slope, it’s crucial to correctly identify and use these coordinates. Misplacing any of these values might lead to incorrect calculations and results in geometry and algebra.

Slope of a Line

The slope of a line is a number that describes how tilted a line is on a graph. It shows the relationship between two points on a line. It can help in understanding how a line moves, whether it goes upwards, downwards, or stays flat.

The formula for finding the slope (denoted as \(m\)) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\\]In simple terms, the slope tells us "rise over run":

• The numerator \(y_2 - y_1\) represents the "rise", or the vertical change between the two points.
• The denominator \(x_2 - x_1\) represents the "run", or the horizontal change between the two points.

A positive slope means the line rises as you move from left to right, while a negative slope indicates it falls. A zero slope means the line is perfectly horizontal, and an undefined slope means the line is vertical. Overall, the slope helps predict the direction and steepness of a line efficiently.

Point-Slope Method

The point-slope method may seem tricky at first, but it's a powerful algebraic tool used to write the equation of a line. It is especially useful when you know:

• The slope of a line.
• One point that the line passes through.

The point-slope formula is written as:\[y - y_1 = m(x - x_1)\\]Here,

• \(m\) is the slope of the line;
• \((x_1, y_1)\) is a known point on that line.

To use this method, plug the known slope and coordinates into the formula. This will give you a linear equation that represents the line.

The beauty of the point-slope method is its simplicity and direct approach, making it easier for students and professionals to find equations of lines efficiently. It's an essential tool for graphing and analyzing lines in algebra.