Problem 94 - Algebra A Combined Function

Question

Write each sentence as an equation or inequality. Use \(x\) to represent any unknown number. 9.1 times a number equals 4

Step-by-Step Solution

Answer

The equation is \( 9.1x = 4 \).

Step 1: Identify the Unknown

The problem states "a number," which means there is an unknown value. We will represent this unknown number with the variable \( x \).

Step 2: Translate Words to Mathematical Expression

The phrase "9.1 times a number" can be translated to \( 9.1 \times x \). "Equals 4" directly translates to \( = 4 \).

Step 3: Combine into Equation

Combine the expressions from the previous step to form the equation: \( 9.1x = 4 \).

Key Concepts

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators that collectively represent a value or a statement. These are foundational elements in algebra and play a crucial role in translating real-world scenarios into solvable problems.

Consider the phrase "9.1 times a number equals 4" from the original problem. This is a real-world scenario expressed in words. To solve such problems, it's necessary to convert these words into a mathematical form, known as a mathematical expression.

• The number "9.1" is a coefficient, which multiplies the unknown quantity.
• The phrase "a number" indicates the presence of an unknown, pointing towards the need for a variable.
• The term "equals 4" establishes a relationship or equality between the expression and the number 4.

Translating this into mathematical expression results in the equation: \(9.1 \times x = 4\). This compact form allows us to manipulate and solve for the unknown quantity.

Variables in Algebra

Variables are symbols used to represent unspecified numbers or values. They're often denoted by letters such as \(x\), \(y\), or \(z\) in algebra.

In the context of problems, variables stand in for numbers we need to find. In our example, the phrase "a number" refers to an unknown value, which we represent by the variable \(x\). This allows us to handle and solve problems without knowing every detail from the start.

• Variables are placeholders for unknowns.
• They're essential for writing general formulas and expressions.
• By assigning a variable, problems can be solved step by step.

Algebraic expressions become much clearer when we substitute these variables. In our case, using \(x\) makes it possible to write the equation \(9.1 \times x = 4\), simplifying the task of finding the unknown number.

Solving Equations

The process of solving equations involves finding the value of the variable that makes the equation true. Once an equation is set up, as in our example \(9.1x = 4\), the next step is finding the number that \(x\) represents.

To solve \(9.1x = 4\), we isolate the variable \(x\). This is done by performing inverse operations, which balances both sides of the equation while keeping the equality intact.

• Perform the opposite action of the current operator. Since \(x\) is multiplied by 9.1, divide both sides by 9.1.
• This simplifies to \(x = \frac{4}{9.1}\).
• Calculating this gives \(x \approx 0.43956\).

Thus, solving equations is about isolating the variable using precise mathematical operations. This process is vital in determining unknown values that fulfill the original word problem.