Problem 87 - Algebra A Combined Function

Question

Find the slope of the line through the given points. $$ (-3.8,1.2) \text { and }(-2.2,4.5) $$

Step-by-Step Solution

Answer

The slope of the line is 2.0625.

Step 1: Understand the Slope Formula

The slope of a line through two points \(x_1, y_1\) and \(x_2, y_2\) is calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). The difference in the y-coordinates is in the numerator and the difference in the x-coordinates is in the denominator.

Step 2: Substitute the Given Points into the Formula

Using the points \((-3.8, 1.2)\) and \((-2.2, 4.5)\), substitute them into the slope formula: \( m = \frac{4.5 - 1.2}{-2.2 - (-3.8)} \).

Step 3: Calculate the Numerator

Subtract the y-coordinates: \( 4.5 - 1.2 = 3.3 \).

Step 4: Calculate the Denominator

Subtract the x-coordinates: \( -2.2 - (-3.8) \) is equivalent to \( -2.2 + 3.8 = 1.6 \).

Step 5: Divide the Numerator by the Denominator

Use the values from the previous steps to find the slope: \( m = \frac{3.3}{1.6} = 2.0625 \).

Key Concepts

Slope Formula

In mathematics, the slope of a line is a measure of its steepness or incline. It tells us how much a line goes up or down for a given distance along the horizontal axis. The slope is often represented by the letter 'm'. The formula to find the slope between two points \((x_1, y_1) \) and \((x_2, y_2)\) is:

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

Here, \(x_1\) and \(y_1\) are the coordinates of the first point, and \((x_2\) and \(y_2\) are the coordinates of the second point. The difference in the y-values \(y_2 - y_1\) shows the vertical change, while the difference in the x-values \(x_2 - x_1\) represents the horizontal change.
The slope allows you to understand the direction and rate of change between two points on a line. If the slope is positive, the line ascends as it moves from left to right. Conversely, a negative slope indicates the line descends. If the slope turns out to be zero, this means the line is perfectly horizontal.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometric figures using a coordinate system. This system allows you to readily identify points, lines, and shapes in a two-dimensional or three-dimensional space. When you work with coordinate geometry, points are often labeled with coordinates \(x, y\), representing their position in a plane.
The slope formula is a crucial part of coordinate geometry. It involves finding out the relationship between two points on the Cartesian plane, as seen in the exercise where the slope of a line connecting \((-3.8, 1.2)\) and \((-2.2, 4.5)\) was calculated.

• Understanding how points relate to each other on a plane, helps in plotting, determining line equations, and solving intersection problems.
• Coordinates give precise positions that are essential for solving distance measurements, midpoint calculations, and other analytical tasks.

Thus, mastering coordinate geometry provides you with the foundational tools needed for deeper mathematical explorations and practical uses.

Mathematics Problem-Solving

Mathematics problem-solving is a key skill that is cultivated through practice and understanding of concepts like the slope formula and coordinate geometry. To successfully solve math problems, follow these basic steps:

• **Understand the Problem:** Read the question carefully, and ensure you comprehend what is being asked. Note down any given data such as coordinates or equations.
• **Plan Your Approach:** Determine which formulas or methods are applicable, like the slope formula in our example.
• **Execute the Plan:** Perform the calculations, ensuring each step follows logically from the last. Substitute numbers accurately to avoid snafus.
• **Check Your Work:** After reaching a solution, verify your answer to ensure it makes sense and covers all parts of the question.

By employing a systematic approach, students can enhance their problem-solving abilities, making the process more efficient and less error-prone. Additionally, practicing problems involving slopes or coordinates fosters a deeper understanding and a greater level of comfort with the concepts, paving the way for more advanced mathematical challenges.