Problem 35 - Fundamentals of Mathematics

Question

Translate each phrase or sentence to a mathematical expression or equation. Five more than some number is three more than four times the number.

Step-by-Step Solution

Answer

The translation is \( x + 5 = 4x + 3 \) and \( x = \frac{2}{3} \).

Step 1: Identify the Variables

Let's define the unknown quantity mentioned in the problem. Let the variable \( x \) represent "some number." This will help us form the mathematical expressions.

Step 2: Translate the First Part of the Sentence

Identify the first part of the sentence: "Five more than some number." This translates to \( x + 5 \), because you take the number \( x \) and add 5 to it.

Step 3: Translate the Second Part of the Sentence

Identify the second part of the sentence: "three more than four times the number." In mathematical terms, this translates to \( 4x + 3 \). Here, you multiply the number \( x \) by 4, and then add 3.

Step 4: Formulate the Equation

Since the sentence indicates that the two parts are equal, we equate the expressions from Steps 2 and 3. This gives us the equation: \( x + 5 = 4x + 3 \).

Step 5: Solve the Equation

Rearrange the equation to solve for \( x \). First, subtract \( x \) from both sides to get: \[ 5 = 3x + 3 \]Then, subtract 3 from both sides to obtain:\[ 2 = 3x \]Finally, divide both sides by 3 to solve for \( x \):\[ x = \frac{2}{3} \]

Key Concepts

Variables

In the world of algebra, variables are symbols used to represent unknown or changeable quantities. They are often denoted by letters like \( x \), \( y \), or \( z \). Understanding variables is crucial because they allow us to generalize mathematical expressions and equations. For instance:

• Variables can stand for numbers we don't yet know, such as in the exercise where \( x \) represents 'some number.'
• They can vary or change depending on the operations performed on them.
• Variables allow us to form equations and expressions, making it easier to solve problems systematically.

When working with variables, keep in mind that they are placeholders. You can manipulate them using basic mathematical operations such as addition, subtraction, multiplication, or division to find their value.

Equations

Equations are mathematical statements that assert the equality of two expressions. They typically involve variables and constants connected by mathematical operations. The word 'equation' comes from 'equal,' highlighting that both sides of the equation balance or are equal. Some key points include:

• Equations can model real-life situations, helping us find unknown values by establishing relationships between known and unknown quantities.
• In our example, "Five more than some number is three more than four times the number," translates into the equation \( x + 5 = 4x + 3 \).
• The equal sign (=) forms the core of any equation, indicating that the expressions on both its sides carry the same value.

Understanding equations is pivotal, as they form the backbone of algebra and many other mathematical concepts. They allow us to set up problems and solve for unknown variables.

Solving Equations

Solving equations is the process of finding the value of the variable that makes the equation true. This involves a series of logical steps aimed at isolating the variable on one side of the equation. Here's a simple breakdown:

• Firstly, simplify the equation if it contains like terms on the same side.
• Rearrange the equation to isolate the variable. This often involves adding or subtracting terms from both sides of the equation.
• Divide or multiply as necessary to solve for the variable.

For the equation \( x + 5 = 4x + 3 \):

• We start by subtracting \( x \) from both sides, simplifying it to \( 5 = 3x + 3 \).
• Then, we subtract 3 from both sides, resulting in \( 2 = 3x \).
• Finally, divide both sides by 3 to isolate \( x \), giving us \( x = \frac{2}{3} \).

Solving equations involves carefully following these steps to ensure the variable is left alone and its value is accurately determined. It teaches us a methodical approach to problem-solving, which is essential in mathematics.