Problem 1 - Elementary Algebra

Question

Find the domain of each of the following equations. Assume that the independent variable is the variable that appears in the expression on the right side of the \("="\) sign. $$ y=5 x+10 $$

Step-by-Step Solution

Answer

Answer: The domain of the given equation is all real numbers, represented by the interval \((-\infty, \infty)\).

Step 1: Identify the equation

The given equation is: $$ y = 5x +10 $$ This is a linear equation, and there are no restrictions on the input values for the independent variable \(x\).

Step 2: Determine the domain

Since there are no restrictions for the independent variable \(x\) in the linear equation, the domain includes all real numbers. We can write it in interval notation as: $$ \text{Domain}: (-\infty, \infty) $$ In conclusion, the domain of the given equation \(y=5x+10\) is all real numbers, represented by the interval \((-\infty, \infty)\).

Key Concepts

Linear Equations

A linear equation is one of the fundamental concepts in algebra. It represents a straight line when plotted on a graph and has the general form of (y = mx + b), where m stands for the slope of the line, and b is the y-intercept, which is the point where the line intersects the y-axis. The x represents the independent variable, and y is the dependent variable.
Within linear equations, the value of x can be any real number, which is why the domain is all real numbers. The simplicity of linear equations makes them incredibly useful for both basic algebra and real-world applications, like calculating expenses over time or determining speed.

• No Restrictions on x: In the equation y = 5x + 10, x can take any value, illustrating that the domain of x is unrestricted for linear equations.
• Real-World Relationships: These equations are often used to find relationships between variables in real-life situations, like converting temperatures or calculating distances.

Independent Variable

In the context of functions and equations, an independent variable is the input value, often denoted by x, that can be freely selected without any constraints from the equation itself. It’s the variable upon which the outcome depends, determining the resulting value of the dependent variable (usually y).
In our equation (y = 5x + 10), x is the independent variable – you can choose any value for x to predict or determine the corresponding y. This is powerful because it allows for prediction or calculation of outcomes based on various scenarios.

• Choice Freedom: The independent variable can be chosen from all real numbers, highlighting its freedom within linear equations.
• Critical for Functions: Recognizing the independent variable is essential to understanding how functions work, especially when graphing them or finding the domain.

Interval Notation

Interval notation is a mathematical way to describe a set of numbers along a continuum. It’s particularly useful for expressing domains and ranges of functions. The notation uses parentheses '()' to indicate that an endpoint is not included, called an open interval, and square brackets '[]' to indicate inclusion, known as a closed interval.
In our solution, we expressed the domain as (-e(), e())), which reads as 'all numbers from negative infinity to positive infinity'. This means that any real number can be input into our linear equation, reflecting the fact that there are no limits on the value of the independent variable.

• Fully Inclusive: The open interval (-e(), e()) is used because we never actually reach infinity, so it’s always considered open.
• Describing Domains: Interval notation is an efficient way to concisely express the domain of an equation or function.