Problem 1 - Introductory and Intermediate Algebra for College Students 4th

Question

Identify the linear equations in one variable. $$x-9=13$$

Step-by-Step Solution

Answer

Yes, the given equation \(x - 9 = 13\) is a linear equation in one variable.

Step 1: Identify the Variable in the Equation

Look at the equation \(x - 9 = 13\). The variable in this equation is 'x'.

Step 2: Check the Power of the Variable

In the equation, the power of the variable 'x' is 1 as it's written in the simplest form, without any exponents.

Step 3: Conclude Whether the equation is Linear or not

Since the given equation has only one variable 'x' and the power of 'x' is 1, this equation is a linear equation in one variable.

Key Concepts

Solving Linear Equations

Understanding the process of solving linear equations is fundamental in algebra. It involves finding the value of the variable that makes the equation true. When we have an equation like \(x - 9 = 13\), our target is to isolate the variable, 'x', on one side of the equation.

First, look at the equation and perform the necessary operation to both sides to keep the equation balanced - in this case, add 9 to both sides to counteract the subtraction of 9 from 'x'. This leads us to \(x - 9 + 9 = 13 + 9\), simplifying to \(x = 22\).

The solution to the equation is the value of 'x' that balances the equation, hence for \(x - 9 = 13\), the solution is \(x = 22\). Learning to accurately perform these operations is crucial as it forms the basis for more complex algebraic concepts.

Variables in Algebra

In algebra, variables are used to represent unknown values. They are usually denoted by letters, such as 'x', 'y', or 'z'. When we see an equation like \(x - 9 = 13\), 'x' is the variable. It stands for a number that we are trying to find. Identifying the variable is the first step in solving an equation.

In the context of linear equations, it's essential to recognize that the variable should appear to the first power, which means there are no exponents other than 1. For example, in our exercise, the variable 'x' is understood to be to the power of 1, even though the exponent is not explicitly written. Recognizing the variable and its power is integral to determining the type of equation you're working with and applying the appropriate method to solve it.

Properties of Linear Equations

Linear equations in one variable exhibit certain properties which are crucial to understanding their behavior. A linear equation will have the following properties:

• Additive Property: You can add or subtract the same number from both sides of an equation without affecting the solution.
• Multiplicative Property: You can multiply or divide both sides by the same nonzero number and the solution will remain valid.
• Linear Equations have One Solution: Typically, a linear equation in one variable has a single unique solution.
• Variable's Power of One: In a linear equation, the variable will always be to the first power.

Understanding these properties helps in identifying and solving linear equations accurately. Together, they form a set of rules that govern the structure and solutions of linear equations, making it easier to approach and solve them systematically.