Problem 25 - College Algebra

Question

For the following exercises, find the slope of the line that passes through the two given points. $$ (1,5) \text { and }(4,11) $$

Step-by-Step Solution

Answer

The slope of the line is 2.

Step 1: Understanding the Slope Formula

The slope of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1}.\) This formula calculates the rate at which \(y\) changes with respect to \(x\).

Step 2: Identify the Coordinates

Identify the coordinates of the given points. Here, the first point \( (1,5) \) has \(x_1 = 1\) and \(y_1 = 5\). The second point \( (4,11) \) has \(x_2 = 4\) and \(y_2 = 11\).

Step 3: Substitute into the Formula

Substitute the coordinates into the slope formula: \( m = \frac{11 - 5}{4 - 1}.\) This will help us find the slope.

Step 4: Simplify the Expression

Simplify the expression to find the slope. Calculate \(11 - 5 = 6\) and \(4 - 1 = 3\). Thus, the slope \(m\) becomes \(\frac{6}{3}\).

Step 5: Calculate the Slope

Divide the numbers obtained in the previous step: \(\frac{6}{3} = 2\). This gives the slope of the line.

Key Concepts

Slope Formula

The slope of a line is a measure of its steepness and direction. It is determined by the change in the vertical direction compared to the change in the horizontal direction. The most common way to find this is by using the slope formula.The slope formula is written as:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where:

• \((x_1, y_1)\) and \((x_2, y_2)\) are any two distinct points on the line
• \(y_2 - y_1\) represents the change in the \(y\)-coordinate, often referred to as the rise
• \(x_2 - x_1\) represents the change in the \(x\)-coordinate, known as the run

The slope \(m\) tells us how much \(y\) increases or decreases when \(x\) increases by 1 unit. A positive slope means the line rises as it moves from left to right. A negative slope signifies that the line falls. Understanding this formula is fundamental in analyzing linear relationships.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometry using a coordinate system. This involves placing every geometric figure in a plane—using a pair of numerical values called coordinates. These coordinates help in visually understanding and analyzing geometric properties.In coordinate geometry, a point is defined by an ordered pair \((x, y)\). For instance, the point \((1, 5)\) describes a location where the \(x\) coordinate is 1 and the \(y\) coordinate is 5. In our case, it is essential to identify these values accurately to substitute into the slope formula.Using the coordinates you've identified, you can apply algebraic methods like the slope formula to solve many problems related to lines on a graph. It connects both algebra and geometry, showing not just where lines lie but also their properties, such as direction and steepness.

Rate of Change

The rate of change is a critical concept in understanding slopes. When you calculate the slope of a line, you essentially determine the rate at which one variable changes in relation to another.Consider the two points \((1, 5)\) and \((4, 11)\). By calculating their slope \(m = \frac{11 - 5}{4 - 1} = 2\), you determine that for every 1 unit increase in \(x\), \(y\) increases by 2 units.The rate of change helps decipher how variables affect each other. It's widely used not only in math but also in physics, economics, and other fields where understanding relationships between variables is crucial. Recognizing this rate allows predictions and interpretations of data trends, making it a powerful tool in both academic and real-world applications.