Problem 26 - College Algebra

Question

For the following exercises, find the slope of the line that passes through the two given points. $$ (-1,4) \text { and }(5,2) $$

Step-by-Step Solution

Answer

The slope of the line is \(-\frac{1}{3}\).

Step 1: Identify Points

We are given two points: \((-1, 4)\) and \((5, 2)\). These points are in the form of \((x_1, y_1)\) and \((x_2, y_2)\). Here, \(x_1 = -1\), \(y_1 = 4\), \(x_2 = 5\), and \(y_2 = 2\).

Step 2: Use the Slope Formula

The formula to find the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substitute the values from Step 1 into this formula.

Step 3: Substitute Values into the Formula

Plug the values \(y_2 = 2\), \(y_1 = 4\), \(x_2 = 5\), and \(x_1 = -1\) into the slope formula: \[ m = \frac{2 - 4}{5 - (-1)} \]

Step 4: Simplify the Slope Expression

Calculate the difference in the numerator: \(2 - 4 = -2\). Calculate the difference in the denominator: \(5 - (-1) = 5 + 1 = 6\). Thus, the slope \(m\) is: \[ m = \frac{-2}{6} \]

Step 5: Simplify the Slope Fraction

Simplify the fraction \(\frac{-2}{6}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2: \[ m = \frac{-2 \div 2}{6 \div 2} = \frac{-1}{3} \] Therefore, the slope of the line is \(-\frac{1}{3}\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a fascinating branch of mathematics that uses the principles of algebra and geometry together. It deals primarily with the study of geometric figures through a coordinate system. In coordinate geometry, points are defined using pairs of numbers, such as \(x, y\).
These coordinates allow us to describe the location of a point in a plane, which can help solve geometric problems more easily. For example, by knowing the coordinates of two points, we can determine several elements, such as the line connecting them and its characteristics, like the slope. The slope is a crucial concept because it gives us the direction and steepness of a line.
Understanding coordinate geometry is essential for tackling problems involving lines, shapes, and curves. It forms the foundation for many fields, such as physics and engineering. This knowledge also transitions into other mathematical fields, aiding in solving complex equations and systems.

Two-Point Formula

The two-point formula is a straightforward method to calculate the slope of a line that intersects two distinct points in a coordinate plane. Slope, denoted by \(m\), is essentially the 'rise' over the 'run', representing how steep or flat a line is. To find the slope \(m\) using the two-point formula, you only need the coordinates of the two points.

• Let's say you have two points, \( (x_1, y_1) \) and \( (x_2, y_2) \).
• The formula for slope is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

This formula illustrates the change in the \(y\)-coordinate divided by the change in the \(x\)-coordinate, representing how the y-values change relative to the x-values. Why is it useful? Because with just two pairs of coordinates, you can define the characteristic slope of the line, which is pivotal for graph interpretation and other advanced calculations.
When working through such formulas, paying attention to signs and correctly substituting values are key skills. Misplacing a negative or a parenthesis may lead to incorrect outcomes. Practicing using examples like this exercise is an excellent springboard towards algebraic proficiency.

Simplifying Fractions

Simplifying fractions is a handy skill in mathematics, making equations simpler and results more digestible. It involves reducing fractions to their lowest terms by dividing the numerator and the denominator by their greatest common divisor (GCD). This process helps us understand the underlying number or ratio more clearly.
For the slope calculation, we found the slope to originally be \[ \frac{-2}{6} \].

• Identify the greatest common divisor of 2 and 6, which is 2.
• Divide both the numerator (-2) and the denominator (6) by 2.
• This yields the simplified slope: \[ \frac{-2 \div 2}{6 \div 2} = \frac{-1}{3} \]

Reducing fractions makes data, whether it is the slope of a line or a statistical value, much neater and easier to work with. Simplified terms are also easier to communicate, which helps in collaborative or educational settings. Always ensure fractions are in their simplest form to maintain clarity and precision in mathematics.