Problem 43 - Elementary Algebra

Question

Translate the following sentence to a mathematical equation. Three less than an unknown number is multiplied by negative four. The result is two more than the original unknown number.

Step-by-Step Solution

Answer

Question: Translate the sentence "Three less than the unknown number, multiplied by negative four, is two more than the original unknown number" into a mathematical equation. Answer: -4(x - 3) = x + 2

Step 1: Identify the Unknown Number

Let the unknown number be x.

Step 2: Translate "Three Less Than the Unknown Number"

This means we should subtract 3 from the unknown number x: (x - 3).

Step 3: Translate "Multiplied by Negative Four"

Multiply the expression from the previous step by -4: -4(x - 3).

Step 4: Translate "The Result is Two More Than the Original Unknown Number"

This means the result of the multiplication should be equal to x plus 2: -4(x - 3) = x + 2.

Step 5: Write the Final Equation

The final equation, representing the given sentence, is: -4(x - 3) = x + 2.

Key Concepts

Algebraic Expressions

When we encounter word problems in mathematics, the first step is often to convert the words into algebraic expressions. An algebraic expression is a combination of numbers, variables, and mathematical operations that represents a particular idea or quantity. For instance, when a word problem mentions 'an unknown number', we introduce a variable, like 'x', to represent this number. If 'three less than an unknown number' is described, the corresponding algebraic expression would be (x - 3).

Algebraic expressions act as the building blocks for creating equations that can solve word problems. Understanding how to craft these expressions from verbal descriptions is crucial as it sets the foundation for equation solving, a key aspect of algebra.

Equation Solving

Once we have translated our word problem into an algebraic equation, the next step is to solve it. Solving an equation means finding the value of the variable that makes the equation true. Let's consider the equation from our original exercise: -4(x - 3) = x + 2. To solve it, we follow systematic steps to isolate the variable.

Combining Like Terms

First, we simplify both sides of the equation by expanding and combining like terms, which can simplify the equation to a more solvable form. In some cases, this might involve moving all the variables to one side and the constants to the other.

Isolating the Variable

Next, we isolate the variable by performing operations to get the variable on one side of the equation, typically ending up with something that looks like 'x = (some number)'.

Checking the Solution

Finally, we verify the solution by substituting the value back into the original equation to ensure it holds true. Each of these steps requires the understanding of algebraic principles and operation rules.

Variable Representation

In algebra, a variable is a symbol, usually a letter, that represents an unknown value. In the context of our exercise, the 'unknown number' is represented by the variable 'x'. The beauty of using variables is the generality they provide—'x' can be any number until we impose certain conditions on it through an equation.

Variables allow for versatility in mathematical expressions and equations. They can stand in for specific numbers we don't yet know, or they can represent a range of possible values. In word problems in particular, variables help us transition from the ambiguity of verbal language to the preciseness of mathematical language, making it possible to perform calculations that lead to solutions.