Problem 14 - Elementary Algebra

Question

In the following problems, solve each of the conditional equations. $$ 3 x=93 $$

Step-by-Step Solution

Answer

Answer: x = 31

Step 1: 1. Given equation

The given equation is $$ 3x=93 $$

Step 2: 2. Divide both sides by 3

To isolate x, divide both sides of the equation by 3: $$ \frac{3x}{3} = \frac{93}{3} $$

Step 3: 3. Simplify the equation

Simplify the equation by carrying out the division: $$ x = 31 $$

Step 4: 4. State the solution

The solution is: $$ x = 31 $$

Key Concepts

Understanding Conditional Equations

In mathematics, a conditional equation is an equation that is true only under certain conditions or for specific values of the variable involved. This makes them different from identities, which are true for all values. In solving a conditional equation like \(3x = 93\), our goal is to find the particular value of \(x\) that makes the equation true. This concept is fundamental in algebra as it helps us pinpoint the exact value or range of values that satisfy the equation.

Conditional equations are a crucial part of learning how to solve for unknowns. It's important to remember that not all equations will have solutions, and those that do may have one or many solutions. In this case, since we are dealing with a linear equation, there will typically be one unique solution.

Exploring Division in Algebra

Division in algebra is a process used to simplify equations and expressions. It involves dividing both sides of an equation by the same nonzero number to maintain equality. This method is key when you need to solve for a variable, as it can help isolate the variable on one side of the equation.

For example, in our equation \(3x = 93\), division is used to solve for \(x\). By dividing both sides by 3, we reduce the equation to \(x = 31\). Here are some reasons why division is important in algebra:

• It helps to simplify complex equations.
• It is used to find a variable's value in linear equations.
• It maintains equality, which is necessary for solving equations correctly.

Remember, when dividing in algebra, always divide each term by the same number to keep the equation balanced.

Isolating the Variable

Isolating the variable is a process in algebra where you aim to get the variable alone on one side of the equation. This makes it much easier to see what the variable equals and is typically the final step in solving an equation.

In our example of \(3x = 93\), we isolated \(x\) by dividing both sides of the equation by the coefficient of \(x\), which is 3. As a result, \(x\) ends up being equal to 31. Here are some key points about isolating variables:

• Helps in determining the specific value of a variable quickly.
• Often involves using inverse operations like division, subtraction, or square roots.
• The goal is to have the variable by itself on one side of the equation.

Successfully isolating the variable is essential for correctly solving equations and understanding the relationship between different parts of an equation.