Problem 6 - 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. The product of 8 and a number is \(272 .\) Find the number

Step-by-Step Solution

Answer

The number is 34

Step 1: Write down the equation

The product of 8 and a number is equal to 272, this can be turned into the equation \(8 * x = 272\).

Step 2: Solve the equation

To solve for \(x\), divide both sides of the equation by 8: \(x = 272 / 8\).

Step 3: Simplify to find the final answer

By performing the division in Step 2, we find that \(x = 34\).

Key Concepts

Equation Solving

Solving equations is a fundamental aspect of algebra that involves finding the unknown variable. The main goal here is to isolate the variable on one side of the equation. This means you'll perform various operations, while keeping the equation balanced. In our exercise, the equation given was:- The product of 8 and a number equals 272.From there, we derived the equation: \[ 8x = 272 \].To solve any equation effectively:

• Understand the problem. Identify key information and translate it into mathematical expressions.
• Simplify the expression if needed. Remove brackets, combine like terms, etc.
• Perform inverse operations to isolate the unknown variable. This includes operations like addition, subtraction, multiplication, or division.

In this case, we used division as the inverse operation of multiplication to isolate \(x\). Importantly, each step should aim to bring the equation closer to expressing \(x\) as the subject of the formula.

Multiplication in Algebra

Multiplication is a fundamental operation in algebra that is frequently used to combine numbers or variables. In algebra, multiplication is often used to model problems and express relationships. In our exercise, we had the expression 8 times a number, represented by:- The equation: \( 8x = 272 \).Here are some key pointers about multiplication in algebra:

• **Using variables**: Variables like \(x\) allow us to represent and manipulate unknown numbers.
• **Coefficients**: The number multiplied by a variable is called a coefficient. In this case, 8 is the coefficient of \(x\).
• **Commutativity**: Multiplication is commutative, meaning that \( ab = ba \). This property allows flexibility in the order of operations.
• **Factors**: Numbers or variables being multiplied are called factors. 8 and \(x\) are factors of 272 in this equation.

In equations involving multiplication, the goal is often to "undo" the multiplication to solve for the variable. This usually involves using the inverse operation, division, which leads us to our next section.

Division in Algebra

Division in algebra is the inverse operation of multiplication and is often used to solve equations involving multiplication. It helps to "undo" multiplication, thereby simplifying and solving equations. Let's look at how division was used in our exercise. Given:- The equation: \( 8x = 272 \),we needed to find \(x\), the unknown number. Here’s the step-by-step on using division:

• **Identify the operation to be undone**: Since 8 is multiplied by \(x\), we use division to undo this multiplication.
• **Apply division to both sides**: Divide both sides of the equation by the same number (8 in this case) to maintain balance. \[x = \frac{272}{8}\]
• **Simplifying**: Perform the division, which in this exercise simplifies to \(x = 34\).

In algebra, division is not only key to isolating variables but also in simplifying expressions and finding solutions. Always remember to perform operations on both sides of the equation to keep it balanced. This ensures the equality holds and your solution remains valid.