Problem 20 - 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. If the quotient of three times a number and four is decreased by three, the result is nine. Find the number.

Step-by-Step Solution

Answer

The number \(x\) represented in the problem statement is 16.

Step 1: Translate the Problem into an Equation

The problem statement can be translated into the following equation: \(3x/4 - 3 = 9\). This equation signifies that three times a number (\(3x\)) divided by four and decreased by three equals nine.

Step 2: Simplify the Equation

The equation \((3x/4) - 3 = 9\) can be simplified by adding 3 to both sides to isolate the fraction on one side. In doing so, we get \(3x/4 = 12\).

Step 3: Solve for \(x\)

Multiply both sides of the equation by 4, to get \(3x = 48\). Divide both sides by 3 to solve for \(x\), yielding \(x=16\).

Key Concepts

Quotient

In mathematics, a **quotient** is the result of a division problem. When we want to find the quotient, we divide one number by another. In our exercise, the term "quotient" refers to the part of the problem that involves dividing three times a number by four, expressed mathematically as \( \frac{3x}{4} \). Here, \(x\) represents the unknown number we need to solve for, and "three times a number" translates to \(3x\). Then, this expression \(3x\) is divided by 4 to get the quotient \(\frac{3x}{4}\). Understanding this concept helps relate abstract mathematical phrases to precise mathematical operations.

Whenever you encounter problems involving quotients, remember to:

• Identify the dividend (the number being divided).
• Identify the divisor (the number by which you divide).
• Write the division expression accurately to form the quotient.

Knowing how to correctly interpret quotients is crucial in setting up the correct equations in algebraic problems.

Isolating Variables

**Isolating variables** is a fundamental skill in solving algebraic equations. It involves manipulating the equation so that the desired variable stands alone on one side of the equation. In our step-by-step solution, isolating the variable \(x\) begins with addressing the equation \(\frac{3x}{4} - 3 = 9\). Our goal is to make \(x\) appear by itself, so we perform operations that gradually simplify the equation.

The initial step involves eliminating constants or other terms from the side of the equation with the variable. Here, we first add 3 to both sides to remove the constant \(-3\), which modifies the equation to \(\frac{3x}{4} = 12\).

• This step is vital because it allows us to focus solely on the term containing \(x\).
• Once this is achieved, additional operations can be performed to further simplify and solve for \(x\).

Mastering the technique of isolating variables enables us to systematically approach and solve linear equations with ease.

Solving Linear Equations

**Solving linear equations** involves finding the value of the variable that makes the equation true. In the linear equation from our exercise, \( \frac{3x}{4} = 12\), solving for \(x\) involves a few clear, systematic steps.

First, to eliminate the fraction, multiply both sides of the equation by 4. This step transforms the equation from \(\frac{3x}{4} = 12\) to \(3x = 48\). Multiplying is crucial here because it clears the fraction, simplifying the equation for further steps.
Next, divide both sides by 3 to isolate \(x\):

• This yields \(x = 16\), which is the solution to the equation.
• Each operation maintains the balance of the equation, ensuring the equality still holds.

The logical sequence of multiplying and then dividing helps in arriving precisely at the value of \(x\). Solving linear equations is about maintaining this balance through each operation, leading straightforwardly to the answer.