Problem 84 - Prealgebra

Question

Translate each of the following into an equation, and then solve the equation. The difference of \(x\) and 12 is \(30 .\)

Step-by-Step Solution

Answer

x = 42

Step 1: Identify the components of the problem

The phrase "The difference of \(x\) and 12 is 30" needs to be translated into an equation. "Difference" typically means subtraction. The problem tells us the result of this subtraction is 30.

Step 2: Translate into a mathematical equation

According to the phrase, the difference of \(x\) and 12 can be expressed as \(x - 12\). Since this difference is equal to 30, our equation becomes \(x - 12 = 30\).

Step 3: Solve the equation for x

To solve the equation \(x - 12 = 30\), add 12 to both sides to isolate \(x\). This gives \(x = 30 + 12\).

Step 4: Calculate the value of x

Perform the addition: \(30 + 12 = 42\). Thus, \(x = 42\).

Key Concepts

Equation Translation

Translating phrases into mathematical equations is a crucial skill in algebra. The first step is always to understand the language of mathematics. For instance, the word "difference" indicates subtraction in this context. When you see a phrase like "the difference of \(x\) and 12 is 30," it is prompting you to formulate this into an algebraic equation.

• Phenomena such as "difference," "sum," "product," and "quotient" indicate basic operations: subtraction, addition, multiplication, and division, respectively.
• Labels such as "is," "was," "equals," often mean the operation is equal to a certain value.

In our example, the phrase "difference of \(x\) and 12" translates directly to \(x - 12\), and because the equation states "is 30," you write \(x - 12 = 30\). Mastering this process is vital, as it is the first step in creating a solvable equation.

Mathematical Expressions

A mathematical expression is a combination of numbers, variables, and operations. It does not include an equality sign (\(=\)). In the context of the example, after the translation step, we deal with both an expression and an equation.

• An expression can include constants (like the number 12), variables (such as \(x\)), and operations (for instance, subtraction).
• The equation formed from the expression includes an equality sign that allows us to solve for the variable in question.

In this lesson, translating the verbal expression "the difference of \(x\) and 12" yields the mathematical expression \(x - 12\). This expression becomes part of the equation \(x - 12 = 30\) when equated to a value.

Basic Algebra

Basic algebra involves finding unknown values by performing operations to isolate variables. These operations should maintain equality. Here's how you can solve a straightforward equation, like \(x - 12 = 30\):

• First, isolate the variable \(x\) by adding or subtracting numbers on the same side of the equation.
• For our example, we add 12 to both sides to counteract subtracting 12, restoring balance to the equation.
• Now you have \(x = 30 + 12\).
• Solve the arithmetic to get \(x = 42\).

This approach is foundational in algebra. Once the variable is isolated, you can determine its value through simple calculation. Practicing these steps will enhance your problem-solving skills in more complex algebraic tasks.