Problem 2 - Algebra A Combined Function

Question

Find the slope of the line that passes through the given points. See Examples 1 and 2. $$ (-1,16) \text { and }(3,4) $$

Step-by-Step Solution

Answer

The slope of the line is -3.

Step 1: Understand the Slope Formula

The slope of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \( m \) represents the slope.

Step 2: Identify the Coordinates

Given points are: \((-1, 16)\) as \((x_1, y_1)\) and \((3, 4)\) as \((x_2, y_2)\).Identify the coordinates: \(x_1 = -1, y_1 = 16, x_2 = 3, y_2 = 4\).

Step 3: Substitute Values into the Formula

Plug the coordinates into the slope formula: \[ m = \frac{4 - 16}{3 - (-1)} \].This simplifies to: \[ m = \frac{4 - 16}{3 + 1} \].

Step 4: Perform the Arithmetic

Compute the numerator: \[ 4 - 16 = -12 \].Compute the denominator: \[ 3 + 1 = 4 \].Therefore, the slope \( m \) is: \[ m = \frac{-12}{4} \].

Step 5: Simplify the Slope

Simplify the fraction \( \frac{-12}{4} \) to get:\[ m = -3 \].Thus, the slope of the line is \(-3\).

Key Concepts

Slope formula

The slope of a line is a key concept in mathematics, particularly in algebra and geometry. To find the slope of a line that passes through two given points, we use a simple, yet powerful formula. This is called the slope formula. The formula is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] - Here, \(m\) stands for the slope of the line.- \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points the line passes through.The slope formula helps us understand the steepness or incline of the line connecting these two points. When you calculate the difference in the \( y \)-coordinates, known as the "rise," and divide it by the difference in \( x \)-coordinates, the "run," you get the slope. Understanding this formula is essential for solving many geometry and algebra problems, as it provides a measure of how one variable changes with respect to another.

Coordinate geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This system allows us to define points in a plane using pairs of numbers (coordinates), making it easier to handle geometric problems numerically. In the exercise, we're given two points,

• \((-1, 16)\) and \((3, 4)\).

To find the slope of the line passing through these points, we employ coordinate geometry principles by selecting the correct coordinates for our formula. - Here, \((-1, 16)\) is defined as \( (x_1, y_1) \).- Similarly, \((3, 4)\) is \( (x_2, y_2) \).By using these selected values, we apply them directly into the slope formula to calculate the slope. Coordinate geometry provides a clear framework and precise measurements to work with geometric figures, allowing for more accurate solutions of problems.

Simplifying fractions

Simplifying fractions is an essential skill in mathematics, and it's used to reduce fractions to their simplest form for easier interpretation and further calculations. When we find the slope using the formula, we may end up with a fraction like \( \frac{-12}{4} \).The process to simplify fractions involves:

• Finding the greatest common divisor (GCD) of the numerator and denominator. For \(-12\) and \(4\), the GCD is \(4\).
• Dividing both the numerator and the denominator by the GCD.

Applying this process here:- Divide \(-12\) by \(4\) to get \(-3\).- Divide \(4\) by \(4\) to get \(1\).So, \( \frac{-12}{4} \) simplifies to \( -3 \). Simplifying such fractions is not just an arithmetic step; it helps in understanding the relationship between the numbers more clearly, providing a cleaner view of the math problem at hand.