Problem 10 - Algebra and Trigonometry

Question

Find the slope of the line through P and Q. $$ P(2,-5), Q(-4,3) $$

Step-by-Step Solution

Answer

The slope of the line is \(-\frac{4}{3}\).

Step 1: Understanding the Formula for Slope

To find the slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the formula for the slope: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula helps us find the change in \(y\) divided by the change in \(x\) between the two points.

Step 2: Substituting the Values

We have two points \(P(2, -5)\) and \(Q(-4, 3)\). Let's substitute these into our slope formula: \( m = \frac{3 - (-5)}{-4 - 2} \).

Step 3: Calculating the Slope

Now we calculate each part. First, for the numerator: \(3 - (-5) = 3 + 5 = 8\). Then, for the denominator: \(-4 - 2 = -6\). So, the slope \( m = \frac{8}{-6} \).

Step 4: Simplifying the Slope

The fraction \(\frac{8}{-6}\) can be simplified. We divide both numerator and denominator by their greatest common divisor, which is 2. This gives \(\frac{8 \div 2}{-6 \div 2} = \frac{4}{-3}\). Therefore, the slope of the line is \(-\frac{4}{3}\).

Key Concepts

Calculate Slope

Calculating the slope of a line is a fundamental skill in geometry and algebra. It allows us to understand how steep a line is and the direction it's heading. The slope \[ m \] is defined as the "rise" over "run", or the vertical change divided by the horizontal change between two points.
To calculate the slope when given two points, \( (x_1, y_1) \) and \( (x_2, y_2) \), you use the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula tells you to subtract the y-coordinate of the first point from the y-coordinate of the second point for the numerator. For the denominator, subtract the x-coordinate of the first point from the x-coordinate of the second point. The result gives you the slope. For instance, if you have points \( P(2, -5) \) and \( Q(-4, 3) \), by substituting these into the formula, you get:

• \((y_2 - y_1) = 3 - (-5) = 8\)
• \((x_2 - x_1) = -4 - 2 = -6\)
• Thus, slope \( m = \frac{8}{-6}\)

Understanding how to find the slope can help you analyze and create graphs effectively.

Points and Lines

The relationship between points and lines is at the heart of many algebra and geometry problems. Each point on a coordinate plane is defined by its pair of coordinates, \( (x, y) \), that represent its position along the horizontal (x-axis) and vertical (y-axis).
When you have two points, you can determine a unique straight line that passes through both. This line is described by its slope and the intercept where it crosses the y-axis, though finding the slope comes first.
Slope lets you understand the connection between two points:

• It tells you if a line is increasing or decreasing.
• A positive slope means the line rises as it moves from left to right.
• A negative slope indicates the line descends from left to right.
• A zero slope suggests the line is perfectly horizontal, while undefined suggests a vertical line.

In our original exercise, the points \((2, -5)\) and \((-4, 3)\) are used to calculate the slope and help define the line crossing through them. Being able to describe and draw lines using points is essential for visualizing mathematical concepts.

Simplifying Fractions

Simplifying fractions is an important step in presenting final answers neatly and understandably. When you calculate a slope and get a fraction, it's always good practice to simplify it.

• Find the greatest common divisor (GCD) of the numerator and the denominator. This will help reduce the fraction to its simplest form.
• For example, from our slope calculation, we found \( \frac{8}{-6} \).
• The GCD of 8 and 6 is 2. So, divide both the numerator and the denominator by 2.
• This gives \( \frac{8 \div 2}{-6 \div 2} = \frac{4}{-3} \).

By simplifying, you ensure the fraction is as compact as possible, making it easier to interpret and apply in further mathematical operations or graphs. Additionally, it often helps in checking for errors because a simplified fraction can be more quickly verified. As you simplify, always maintain the sign of the fraction. If the denominator is negative, you often bring the negative sign to the numerator or out front to keep conventions consistent.