Problem 4 - Introductory Algebra for College Students

Question

Find the slope and the \(y\) -intercept of the line with the given equation. $$y=4 x-2$$

Step-by-Step Solution

Answer

The slope of the line is 4, and the y-intercept is -2.

Step 1: Identify the Slope

The slope of the equation \(y = 4x - 2\) is the coefficient of \(x\). In this case, the slope \(m\) is 4.

Step 2: Identify the y-Intercept

The y-intercept \(b\) of the equation \(y = 4x - 2\) is the constant term, which is -2.

Key Concepts

Linear Equations

Linear equations are foundational in algebra and represent lines in a two-dimensional space. These equations take the general form of \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) represents the y-intercept. In simple terms, a linear equation tells you how one variable (typically \(y\)) changes with respect to another variable (\(x\)).
Understanding linear equations is crucial because they describe relationships between variables that are constant and predictable. For example, the equation \(y = 4x - 2\) details that for every one-unit increase in \(x\), \(y\) increases by four units, and when \(x\) is zero, \(y\) will be \(-2\).

Slope of a Line

The slope of a line is a number that describes how steep the line is. Algebraically, it is represented by the letter \(m\) in the linear equation. Slope is calculated by the rise over run between two points on the line, often written as \(m = \frac{\Delta y}{\Delta x}\), where \(\Delta y\) is the change in \(y\) and \(\Delta x\) is the change in \(x\).
For the equation \(y = 4x - 2\), we identify the slope as \(4\). This means the line rises 4 units vertically for every 1 unit it runs horizontally, making it relatively steep. A positive slope like this indicates the line moves upward as one moves to the right along the axis, while a negative slope would indicate a downward trend.

Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis of a graph. On a coordinate plane, this is where \(x = 0\). It's represented by the letter \(b\) in the equation of a line \(y = mx + b\). The y-intercept provides a starting point for graphing the line and for understanding its position relative to the origin.
Looking at the equation \(y = 4x - 2\), the y-intercept \(b\) is \(-2\). This tells us that the line crosses the y-axis below the origin (since it is negative) at the point \((0, -2)\). Knowing the y-intercept is essential for correctly plotting a linear graph as it provides a concrete location to begin the graph.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (such as addition, subtraction, multiplication, or division). These expressions are the building blocks of algebra and are used to describe relationships between different quantities. They do not contain an equals sign, which distinguishes them from equations.
In our exercise, \(4x - 2\) is an algebraic expression that can be part of the linear equation \(y = 4x - 2\). This expression includes a variable \(x\), coefficients (4 and -2), and an operation (subtraction). Proper understanding of algebraic expressions is key to solving and manipulating algebraic equations, allowing for prediction and analysis of various scenarios mathematically.