Problem 61 - Algebra A Combined Function

Question

Find the slope of the line containing each pair of points. (4,-1),(5,-2)

Step-by-Step Solution

Answer

The slope of the line is -1.

Step 1: Identify the Points

The given points are (4, -1) and (5, -2). Label them as \((x_1, y_1) = (4, -1)\) and \((x_2, y_2) = (5, -2)\).

Step 2: Understand the Slope Formula

The formula for finding the slope \(m\) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Step 3: Substitute the Values

Substitute the known values into the slope formula: \(m = \frac{-2 - (-1)}{5 - 4}\).

Step 4: Simplify the Numerator

Calculate the change in \(y\): \(-2 - (-1) = -2 + 1 = -1\).

Step 5: Simplify the Denominator

Calculate the change in \(x\): \(5 - 4 = 1\).

Step 6: Calculate the Slope

Substitute the simplified values into the formula: \(m = \frac{-1}{1} = -1\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebra to explore geometric shapes and relationships. In coordinate geometry, points are defined using coordinate pairs

• The first number represents the x-coordinate and tells you how far along the horizontal axis a point is.
• The second number represents the y-coordinate, indicating the position along the vertical axis.

For example, the point (4, -1) tells us that it is 4 units along the x-axis and -1 unit down the y-axis. This concept is valuable as it allows the precise location of points in the plane and provides a foundation to examine lines, angles, and shapes using algebraic formulas. By understanding the coordinates, we can employ geometric formulas, like the slope formula, to calculate the characteristics of lines that pass through these points.

Slope Formula

The slope of a line in coordinate geometry is a measure of its steepness and direction. It tells us how much the y-coordinate changes for a change in the x-coordinate. The slope formula is given by:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula represents the "rise over run," which is a way of saying how much you move up or down (the rise) for how much you move left or right (the run). In the given problem, once we identify our points as (4, -1) and (5, -2), we substitute into the slope formula:

• The change in y (rise) is found by computing \(-2 - (-1) = -1\).
• The change in x (run) is found by computing \(5 - 4 = 1\).

By putting these changes into the formula, we get a slope of \(-1\), indicating the line is falling.

Linear Equations

Linear equations are mathematical statements of equality involving a linear expression. A linear equation describes a line in the coordinate plane and typically has the following form:\[y = mx + b\]where:

• \(m\) is the slope of the line.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

Understanding how to find the slope from given points is crucial because it allows us to write the equation of the line that passes through those points. Using the slope we found, if we wanted to, we could further determine the y-intercept and fully establish the line's equation. This is the foundation of graphing and analyzing straight lines in coordinate geometry.

Algebra Concepts

Algebra is the study of mathematical symbols and rules to manipulate these symbols. In the current problem, simplifying expressions such as \(-2 - (-1)\) is a basic algebra concept. Simplifying this expression involves changing a subtraction of a negative into an addition, resulting in the simpler expression \(-2 + 1\). This rules of algebra for simplification are essential tools for solving equations accurately. Additionally, understanding how to manipulate expressions in the slope formula helps to determine the slope precisely. Overall, mastering algebra concepts such as manipulation of expressions and equations is vital not only to find slopes but in a broad array of mathematical problems.