Problem 61 - Prealgebra

Question

Translate each sentence below into an equation, then solve the equation. The sum of \(2 x\) and 5 is 19

Step-by-Step Solution

Answer

The value of \(x\) is 7.

Step 1: Translate the Sentence into an Equation

The sentence "The sum of \(2x\) and 5 is 19" can be translated into the equation \(2x + 5 = 19\). Here, "sum" indicates addition, and "is" indicates the equals sign.

Step 2: Isolate the Variable Term

To isolate the term \(2x\), subtract 5 from both sides of the equation: \(2x + 5 - 5 = 19 - 5\). This simplifies to \(2x = 14\).

Step 3: Solve for \(x\)

To solve for \(x\), divide both sides of the equation by 2: \(\frac{2x}{2} = \frac{14}{2}\). This gives \(x = 7\).

Key Concepts

Variable Isolation

Variable isolation is a key step in solving equations. It involves manipulating the equation to get the variable alone on one side. The goal is to have the variable, such as \( x \), by itself so you can determine its value. This often involves using inverse operations. For example, if addition is present, subtraction can be used, and vice versa. The phrase 'get the variable by itself' can often help you visualize what needs to be done.
In our problem, after translating the sentence into an equation, we saw the equation \( 2x + 5 = 19 \). To isolate \( 2x \), we used subtraction. By subtracting 5 from both sides, we kept the equation balanced. This step is crucial because it transforms the equation into \( 2x = 14 \), making it easier to solve for \( x \). Remember, what you do to one side must always be done to the other to maintain balance!

Equation Translation

Equation translation is the vital first step in solving word problems. It involves converting a problem stated in words into a mathematical equation. Think of it like transforming a sentence into math symbols that are much easier to solve. Identifying key phrases in the problem helps guide this translation.

• 'Sum' tells us to add two quantities
• 'Is' often signals the equals sign

In the sentence "The sum of \( 2x \) and 5 is 19," the word 'sum' indicates addition, while 'is' suggests where the equation will equal another number. Translating this sentence, we form the equation \( 2x + 5 = 19 \). Proper translation ensures your equation accurately represents the problem you're trying to solve.

Basic Algebra

Basic algebra involves the fundamental skills needed to solve problems like the given exercise. It focuses on knowing how to manipulate numbers and symbols to find unknown values. Algebra uses operations like addition, subtraction, multiplication, and division to find solutions.
In our problem, once we translated the word problem into an equation and isolated the variable \( 2x = 14 \), we used division to solve for the variable \( x \). By dividing both sides of the equation by 2, we simplify to find that \( x = 7 \). This is a perfect example of applying basic algebra skills to unravel and solve an equation efficiently. Understanding these steps ensures that no matter the complexity of the problem, you can break it down and solve it using these fundamental techniques.