Question

In Problems \(23-28\), find the slope of the line containing the given two points. (2,3) \text { and }(-5,-6)

Short answer

The slope is \(\frac{9}{7}\).

Step-by-step solution

Understand the Formula for Slope

The formula to find the slope (\(m\)) of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). This formula calculates the change in the \(y\)-coordinates divided by the change in the \(x\)-coordinates.

Identify the Points and Variables

Assign the given points to \((x_1, y_1)\) and \((x_2, y_2)\): Point 1 is \((2, 3)\) where \(x_1 = 2\) and \(y_1 = 3\); Point 2 is \((-5, -6)\) where \(x_2 = -5\) and \(y_2 = -6\).

Substitute Values into the Slope Formula

Substitute the values of the points into the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-6 - 3}{-5 - 2}\).

Calculate the Numerator and Denominator

Calculate the change in \(y\)-coordinates: \(-6 - 3 = -9\). Calculate the change in \(x\)-coordinates: \(-5 - 2 = -7\).

Divide the Differences to Find the Slope

Now, put these values in the division: \(m = \frac{-9}{-7}\). Simplify the fraction: \(m = \frac{9}{7}\).

Conclude with the Final Slope

The slope of the line containing the points \((2, 3)\) and \((-5, -6)\) is \(\frac{9}{7}\).

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a fascinating branch of mathematics. It connects algebra and geometry through graphs and equations, providing a more comprehensive understanding of both areas. In coordinate geometry, points are defined by numerical coordinates, usually in the form of (\(x\), \(y\)) in a two-dimensional space.

• These coordinates represent points on a plane. Each point corresponds to a specific location.
• The Cartesian plane is divided into four quadrants by the intersection of the x-axis (horizontal) and y-axis (vertical).
• Lines, curves, and shapes can be represented using equations, making it easier to analyze their properties.

Understanding coordinate geometry is fundamental when solving problems involving lines, slopes, and graphing functions. It helps visualize mathematical concepts, leading to better problem-solving skills.

Finding Slope

Finding the slope of a line is a crucial skill in coordinate geometry. The slope represents how steep the line is and its direction. Knowing the slope helps predict how one variable changes with another in a graph.

To find the slope, given two points \((x_1, y_1)\) and \((x_2, y_2)\), use the slope formula:

• Mathematically, the formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
• The numerator (\(y_2 - y_1\)) gives the change in the vertical direction. This is the change in the y-coordinates.
• The denominator (\(x_2 - x_1\)) represents the change horizontally. This means the change in x-coordinates.

When using the formula, ensure the points are inserted correctly into the formula. The slope will tell you whether the line ascends, descends, or is flat. A positive result means it climbs upward, negative indicates downward, and zero denotes a horizontal line. This simple calculation elucidates the relationship between coordinates in a plane.

Mathematics Problem-Solving

Mathematics problem-solving is about finding practical and logical solutions to mathematical puzzles. It involves understanding the problem, planning an approach, and executing the plan methodically. In problems involving slopes and coordinates, specific strategies enhance understanding and lead to accurate results.

Here's a structured approach to problem-solving in coordinate geometry:

• Understand the Problem: Determine what needs to be found, like a slope between two points in our exercise.
• Gather Information: Identify given numerical data, such as specific point coordinates.
• Choose a Formula or Strategy: Select the appropriate mathematical formula, like the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), and understand how and when to apply it.
• Calculate Carefully: Substitute numbers correctly, execute the arithmetic steps precisely to prevent mistakes.
• Verify the Results: Double-check calculations for accuracy and logic. Review the entire problem context to confirm if the solution accurately addresses the question.

Problem-solving in mathematics builds critical thinking and logical reasoning skills, which are valuable in academic and real-world scenarios.