Problem 16 - Algebra for College Students

Question

Translate each verbal sentence into an equation, using \(x\) as the variable. Then solve the equation. $$ \text { The sum of a number and }-4 \text { is } 18 \text { . Find the number. } $$

Step-by-Step Solution

Answer

x = 22

Step 1: Translate the Verbal Sentence into an Equation

We need to identify the mathematical relationship described in the sentence. The problem states: 'The sum of a number and -4 is 18'. Using the variable x to represent the number, we can translate the sentence into the following equation: \[ x + (-4) = 18 \]

Step 2: Simplify the Equation

Simplify the equation by combining like terms. \[ x - 4 = 18 \]

Step 3: Solve for the Variable

To find the value of x, isolate x on one side of the equation by adding 4 to both sides: \[ x - 4 + 4 = 18 + 4 \]This simplifies to \[ x = 22 \]

Key Concepts

Verbal Sentences to Equations

Understanding how to translate verbal sentences into equations is a fundamental skill in algebra. This process involves identifying key mathematical relationships within a sentence and then expressing them using mathematical symbols. Let's take the example provided: 'The sum of a number and -4 is 18'.
The word 'sum' tells us to add two quantities, here 'a number' (which we will represent with the variable \(/x/\)) and -4. The word 'is' indicates equality, represented by the equals sign (\(/=/\)). Putting this together, we translate the sentence into the equation:
\[ x + (-4) = 18 \] Remember, translating verbal sentences into equations is like converting a language you speak every day into the universal language of mathematics. Always look for words that indicate operations (such as sum, difference, product) and relationships (such as is, less than, greater than).

Simplifying Equations

Once you have the equation, the next step is to simplify it by combining like terms. In our specific example:
\[ x + (-4) = 18 \], we combine the terms involving numbers. Simplifying here is straightforward:
\[ x - 4 = 18 \]
Combining like terms means putting together terms with variables and constants on their own. This simplification makes the equation easier to handle when isolating the variable. Simplifying is crucial because dealing with a simpler equation reduces the risk of mistakes and makes the next steps much easier.

Isolating Variables

Isolating the variable is about getting the unknown variable by itself on one side of the equation so that we can understand its value. Starting from the simplified equation:
\[ x - 4 = 18 \], we need to cancel out the -4 attached to x. We do this by performing the opposite operation; here, adding 4 to both sides of the equation. This keeps the equation balanced:
\[ x - 4 + 4 = 18 + 4 \]
This further simplifies to:
\[ x = 22 \]
Now, the variable x is isolated, and we have found its value. Isolating the variable involves undoing any arithmetic operations that have been performed on the variable. Always remember to perform the same operation on both sides of the equation to maintain balance.