Problem 17 - Elementary Algebra

Question

The formula for finding the slope of a line through any two given points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) is \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

Step-by-Step Solution

Answer

Answer: The slope of the line is 2.

Step 1: Determine the given points

First, let's assume we have two given points to work with. Let's say the points are \(A(2, 3)\) and \(B(4, 7)\). These are represented as \((x_{1}, y_{1})=(2, 3)\) and \((x_{2}, y_{2})=(4, 7)\).

Step 2: Apply the slope formula

Now, we will use the given slope formula \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\). Plug in the coordinates of points A and B into the formula: \(m = \frac{7-3}{4-2}\).

Step 3: Calculate the slope

Simplify the expression in the formula to find the slope of the line: \(m = \frac{4}{2} = 2\). So, the slope of the line through points A and B is 2.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that describe a straight line on a graph. The most common form of a linear equation is the slope-intercept form, written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This equation shows how \(y\) changes with respect to \(x\).
The slope \(m\) indicates how steep the line is; a larger slope means a steeper line, while a smaller slope leads to a flatter line. The y-intercept \(b\) is the point where the line crosses the y-axis.

• If the slope is positive, the line ascends as it moves to the right.
• If the slope is negative, the line descends as it moves to the right.
• A zero slope indicates a horizontal line, and an undefined slope corresponds to a vertical line.

Understanding linear equations is fundamental because they are used to model real-world situations and help solve various mathematical problems.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses coordinates to describe geometric figures. By using a coordinate plane, we can analyze and understand the properties of shapes and their relations, like lines, circles, and other figures.
In the coordinate plane, points are identified using pairs \(x, y\), called coordinates, which depict their position relative to the horizontal x-axis and vertical y-axis. For example, the point \(A(2, 3)\) means it is 2 units right of the origin and 3 units up.
One of the key utilities of coordinate geometry is finding the distance between two points and determining the slope of a line that passes through them. The slope itself, calculated through the formula \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\), is a crucial concept for understanding the angle and direction of the line.
Coordinate geometry serves as a bridge between algebra and geometry by providing methods to prove geometric concepts through algebraic equations and techniques.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It serves as the foundation for writing and solving equations and inequalities. In algebra, letters like \(x\) and \(y\) often represent numbers, and these variables can stand in for unknowns or quantities that vary.
Algebraic skills are essential for solving equations, like the linear equation used in the slope formula. When you solve for the slope \(m\), you apply algebraic techniques to manipulate the expression and isolate the variable representing the slope.
Some basic algebra operations involve:

• Adding or subtracting from both sides of an equation to maintain equality.
• Multiplying or dividing all terms in an equation by the same number, again to maintain balance.
• Simplifying expressions to find solutions more easily.

Algebras' power lies in its universal application, from solving simple linear equations to understanding complex systems. It establishes a language through which mathematical ideas can be communicated, calculated, and understood effectively.