Problem 14 - Introductory and Intermediate Algebra for College Students 4th

Question

Let \(x\) represent the number. Use the given conditions to write an equation. Solve the equation and find the number. A number increased by 12 is four times the number. Find the number.

Step-by-Step Solution

Answer

The number is 4.

Step 1: Setting Up The Equation

Translate the problem statement into an equation. From the problem, 'a number increased by 12 is four times the number' translates into \(x + 12 = 4x\). This is the equation to solve.

Step 2: Subtract \(x\) from both sides

To solve for \(x\), start by getting all terms with \(x\) on one side. Subtract \(x\) from each side of the equation. This gives \(12 = 4x - x\) or \(12 = 3x\).

Step 3: Solving for \(x\)

With the equation \(12 = 3x\), divide both sides by 3 to solve for \(x\). This gives \(x = 12 / 3\) or \(x = 4\).

Key Concepts

Understanding Algebraic Equations

Algebraic equations are the backbone of solving many mathematical problems. They consist of variables, numbers, and operations that are set equal to each other. For instance, the equation from our exercise,

\(x + 12 = 4x\)

is a simple algebraic equation where \(x\) represents the number we're trying to find. Understanding these equations involves recognizing the parts of the equation and how they interact. The left side, \(x + 12\), describes a number increased by 12. The right side, \(4x\), represents four times the number. By comparing the two sides, we set up a relationship that we can manipulate to find the value of \(x\).

When approaching algebraic equations, the main goal is to isolate the variable of interest, in this case, \(x\), to determine its value.

The Art of Variable Isolation

Variable isolation is a crucial technique in solving algebraic equations and is central to unveiling the unknowns the problems contain. It involves rearranging the equation so the variable stands alone on one side of the equals sign. In our exercise, we started with
\(x + 12 = 4x\)

and aimed to isolate \(x\). This was achieved using the fundamental property of equality which states that what you do to one side of the equation, you must do to the other. Subtracting \(x\) from both sides gave us \(12 = 3x\), simplifying the equation and bringing us one step closer to finding the value of \(x\).

The final step in variable isolation involves dividing both sides of the equation by the coefficient of the variable, in this case, 3, giving us the solution \(x = 4\). It's important to perform operations that 'undo' the operations attached to the variable. For multiplication, we divide; for addition, we subtract, and so on.

Translating Word Problems to Equations

Many students stumble on translating word problems into algebraic equations, yet it's a pivotal skill in mathematics. Let's consider the exercise where we were given the statement 'A number increased by 12 is four times the number.' The trick is to identify keywords and turn them into mathematical operations. 'Increased by' suggests addition, so we add 12 to the number \(x\). 'Is' indicates equality, leading us to set up the equal sign. Finally, 'four times the number' translates to multiplying \(x\) by 4.

By assembling these pieces, we develop the equation \(x + 12 = 4x\). Translating words to symbols requires practice and a careful reading of the problem. Therefore, to excel in solving word problems, it's essential to build a strong vocabulary of terms commonly used in mathematics and understand the operations they represent.