Slope Calculation: Points (-4, 5) And (3, -1) Explained

by Alex Johnson 56 views

Finding the slope of a line is a fundamental concept in mathematics, particularly in algebra and geometry. The slope describes the steepness and direction of a line, and it's crucial for understanding linear relationships. In this comprehensive guide, we'll walk you through the process of calculating the slope of a line that passes through two given points: (-4, 5) and (3, -1). We'll break down the formula, provide step-by-step instructions, and offer clear explanations to ensure you grasp this essential concept. Let's dive in and explore how to determine the slope with ease.

Understanding the Slope Formula

To calculate the slope, we first need to understand the slope formula. The slope, often denoted by the variable m, represents the ratio of the vertical change (rise) to the horizontal change (run) between any two points on a line. The formula is expressed as:

m = (y₂ - y₁) / (x₂ - x₁)

Where:

  • (x₁, y₁) are the coordinates of the first point.
  • (x₂, y₂) are the coordinates of the second point.

The slope formula is derived from the concept of rise over run, where the rise is the change in the y-coordinates (vertical change), and the run is the change in the x-coordinates (horizontal change). By dividing the rise by the run, we obtain a numerical value that represents the steepness and direction of the line. A positive slope indicates that the line rises from left to right, while a negative slope indicates that the line falls from left to right. A slope of zero represents a horizontal line, and an undefined slope represents a vertical line. Understanding the components of the slope formula is crucial for accurately calculating the slope of a line given any two points.

Identifying the Coordinates

Before we can apply the slope formula, we need to identify the coordinates of the two given points. In this case, we have the points (-4, 5) and (3, -1). Let's label these points as follows:

  • Point 1: (-4, 5) which means x₁ = -4 and y₁ = 5
  • Point 2: (3, -1) which means x₂ = 3 and y₂ = -1

Correctly identifying the x and y coordinates for each point is a crucial first step in calculating the slope. Mislabeling the coordinates will lead to an incorrect slope calculation. It is helpful to write down the coordinates clearly and label them to avoid confusion. For instance, you can write x₁ = -4, y₁ = 5, x₂ = 3, and y₂ = -1 before substituting these values into the slope formula. This simple step can significantly reduce the chances of making a mistake. Once the coordinates are correctly identified, you can proceed with confidence to the next step, which involves substituting these values into the slope formula and simplifying the expression.

Substituting the Values

Now that we have identified our coordinates, we can substitute them into the slope formula:

m = (y₂ - y₁) / (x₂ - x₁)

Substituting the values we identified:

m = (-1 - 5) / (3 - (-4))

This step is where the actual calculation begins, and it's crucial to be meticulous with the signs and order of operations. Substituting the values correctly ensures that the subsequent steps will lead to the accurate slope. Double-checking the substitution is always a good practice to avoid errors. For example, ensure that the y-coordinate of the second point (-1) is correctly placed as y₂ and the y-coordinate of the first point (5) is placed as y₁. Similarly, verify that the x-coordinates are in their correct positions: x₂ (3) and x₁ (-4). The negative signs in the formula and the coordinates themselves need careful attention to avoid confusion. Once the substitution is complete and verified, the next step is to simplify the expression, which involves performing the subtraction in the numerator and the denominator.

Simplifying the Expression

Next, we simplify the expression by performing the subtractions:

m = (-1 - 5) / (3 - (-4)) m = (-6) / (3 + 4) m = -6 / 7

Simplifying the expression is a critical step in determining the slope. The arithmetic operations, particularly subtraction and dealing with negative signs, need to be handled with care to arrive at the correct result. In the numerator, subtracting 5 from -1 results in -6. In the denominator, subtracting a negative number is equivalent to adding the positive counterpart, so 3 - (-4) becomes 3 + 4, which equals 7. After performing these operations, the slope is expressed as -6 / 7. This fraction represents the slope of the line passing through the given points. It's essential to simplify the fraction to its lowest terms if possible, but in this case, -6 / 7 is already in its simplest form. The simplified expression provides a clear and concise representation of the line's slope, indicating both its steepness and direction.

Final Result: The Slope

Therefore, the slope of the line that passes through the points (-4, 5) and (3, -1) is -6/7. This means that for every 7 units we move to the right along the line, we move 6 units down. The negative sign indicates that the line slopes downward from left to right.

The final result, -6/7, represents the slope of the line passing through the points (-4, 5) and (3, -1). This value provides a concise description of the line's steepness and direction. The negative sign signifies that the line has a downward slope, meaning it descends from left to right. The fraction 6/7 indicates the rate at which the line descends: for every 7 units of horizontal movement (the