Problem 22 - Elementary Algebra

Question

In the following exercises, solve each number word problem. The difference of four times a number and seven is 21 . Find the number.

Step-by-Step Solution

Answer

The number is 7.

Step 1: Understand the Problem

The problem states that the difference of four times a number and seven equals 21. We need to find the unknown number.

Step 2: Assign a Variable

Let the unknown number be represented by the variable \( x \).

Step 3: Set Up the Equation

According to the problem, four times the number minus seven equals 21. This translates to the equation: \( 4x - 7 = 21 \)

Step 4: Isolate the Variable

To solve for \( x \), first add 7 to both sides of the equation to get: \( 4x - 7 + 7 = 21 + 7 \) Simplifying this gives us: \( 4x = 28 \)

Step 5: Solve for the Variable

Now, divide both sides by 4 to isolate \( x \): \( x = \frac{28}{4} \) Simplifying this, we get: \( x = 7 \)

Step 6: Verify the Solution

To verify, substitute \( x = 7 \) back into the original equation: \( 4(7) - 7 = 21 \) Calculating this: \( 28 - 7 = 21 \) This confirms that the solution is correct.

Key Concepts

algebraic equations

Algebraic equations form the backbone of many mathematical problem-solving processes. An algebraic equation is an equation that includes variables, constants, and arithmetic operations. In our example,
the equation is: \(4x - 7 = 21\).
Recognizing this structure helps us see how the parts of the problem fit together. Here's a breakdown:

• The variable, represented by \(x\), stands for unknown quantities we're trying to find.
• Coefficients are constants multiplied by the variable, which in this case is 4.
• Other constants, like -7, alter the quantity in the equation.

By framing a real-world problem as an algebraic equation, we translate everyday language into a form that math can easily solve. This equation translation helps us systematically tackle the problem by applying arithmetic and algebraic rules in a logical sequence.

variable isolation

Variable isolation is the process of rearranging an equation to get the variable on one side by itself. This is done to solve for the variable.
Start with the given equation: \(4x - 7 = 21\)

The goal is to isolate \(x\). Follow these steps:

• Add 7 to both sides to get rid of the -7: \(4x - 7 + 7 = 21 + 7\).
• You're left with: \(4x = 28\).
• Next, divide both sides by 4 to solve for \(x\): \(x = \frac{28}{4}\).
• The result is: \(x = 7\).

Notice how each step brings you closer to isolating \(x\). Addition and subtraction help remove constants, while division and multiplication adjust the coefficient of the variable. Through systematic operations, you isolate the variable, enabling you to determine its value.

word problems

Solving word problems often feels like a puzzle, where you need to extract the algebraic equation from a story. Here, let's decode the word problem step-by-step:

• Identify the unknown quantity: In this scenario, it's the number we need to find.
• Assign a variable to this unknown: Let's call it \(x\).
• Translate the words into an equation: “Four times a number” is \(4x\), and “the difference of four times a number and seven” translates to \(4x - 7\). This equals 21, so we write: \(4x - 7 = 21\).

By translating the words into a math equation, you can use algebraic methods to find the solution. Practice on different word problems to get better at this translation process.
Putting it all together, understanding algebraic equations, how to isolate variables, and translating word problems into these forms will equip you with powerful tools for solving diverse mathematical challenges.