Problem 48 - Fundamentals of Mathematics

Question

Translate each phrase or sentence to a mathematical expression or equation. A number is subtracted from nine. This result is then increased by one. The result is eight more than three times the number.

Step-by-Step Solution

Answer

The number is \(\frac{1}{2}\).

Step 1: Define the Variable

Let's define the variable to help translate the problem into a mathematical equation. Let \( x \) be the number we are trying to find.

Step 2: Translate the First Phrase

The phrase "a number is subtracted from nine" can be expressed as \( 9 - x \).

Step 3: Translate the Second Phrase

The second part of the problem states "this result is then increased by one." This translates to \((9 - x) + 1\), which simplifies to \(10 - x\).

Step 4: Translate the Third Phrase

We are told that the result is "eight more than three times the number." This is expressed as \(3x + 8\).

Step 5: Formulate the Equation

Now, form the equation by setting the expression from Step 3 equal to the expression from Step 4: \(10 - x = 3x + 8\).

Step 6: Simplify and Solve the Equation

To solve the equation \(10 - x = 3x + 8\), first add \(x\) to both sides to get \(10 = 4x + 8\). Then subtract 8 from both sides: \(2 = 4x\). Finally, divide by 4: \(x = \frac{1}{2}\).

Key Concepts

Equations

In mathematics, equations are like statements that show two expressions are equal. They are essential tools that help us solve problems in algebra. An equation will usually involve an equal sign, illustrating the balance between two sides. For instance, in the expression \(10 - x = 3x + 8\), the equation indicates that the value on the left side, \(10 - x\), is equal to the expression on the right side, \(3x + 8\).
This balance is a critical aspect of solving equations, as it ensures that we'll find the correct value for the variable that makes both sides equal.

• Equations often arise from problem-solving scenarios where conditions or relationships are defined through equalities.
• They help us represent complex scenarios in a simple mathematical framework.

Understanding and manipulating equations involves several steps, such as simplifying, weighing both sides evenly, and usually isolating the variable to find its value.

Variables

Variables represent unknown numbers or values in mathematical expressions and equations. They are typically denoted by letters like \(x\) or \(y\), helping to convey complex ideas succinctly. In our example, \(x\) is the variable representing the mysterious number we want to find.

• Variables are placeholders in mathematical expressions, making it possible to generalize and solve myriad problems.
• They make it easier to manipulate and solve equations, as we can perform operations to isolate and find their values.

The beauty of using variables is their ability to adapt and fit into various mathematical models, showcasing the versatility of algebra in problem-solving. In our exercise, \(x\) was crucial for translating sentences into mathematical expressions.

Mathematical expressions

Mathematical expressions are combinations of numbers, variables, and operators, such as addition or multiplication, without an equal sign. These expressions allow us to model real-world situations mathematically. In the given problem, expressions such as \(9 - x\) and \(3x + 8\) are used to translate verbal descriptions into numeric form.

• Expressions can involve constants (like 9 or 8), variables (like \(x\)), and mathematical operations (like subtraction).
• They help in understanding and processing mathematical relationships and patterns.

Mathematical expressions build the foundation for forming equations, letting us capture and solve intricate problems. By breaking down statements into parts using expressions, we can construct equations like in the original solution and work toward finding specific answers.