Problem 60 - Introductory Algebra for College Students

Question

Use the given information to write an equation. Let x represent the number described in each exercise. Then solve the equation and find the number. If 23 is subtracted from a number, the result is \(-8 .\) Find the number.

Step-by-Step Solution

Answer

The number from the given condition is 15.

Step 1: Translate the word equation into a mathematical equation

The word equation can be translated into a mathematical equation. The phrase 'a number' can be represented by the variable \(x\), '23 is subtracted from' can be represented by the operator \(-23\). The result being -8 gives us the rest of the equation. So, this translates to: \(x - 23 = -8\)

Step 2: Solve the mathematical equation

We now rearrange the equation to solve for \(x\). Simply, by adding 23 to both sides of the equation to isolate \(x\) gives us the value of \(x\): \(x = -8 + 23\).

Step 3: Simplify the equation

Finally, we can simplify the equation by conducting the operation on the right side of the equation, giving us: \(x = 15\). So, the number from the given condition is 15.

Key Concepts

Understanding Equations

Equations are fundamental components of algebra. They are mathematical statements where two expressions are set equal to each other, symbolized by the equal sign \(=\). In the exercise, we dealt with a simple equation \(x - 23 = -8\). Here, the value \(x\) represents the unknown number we are trying to find. Equations can appear as simple as the one given, or they might become complex as you delve deeper into algebra.

Typically, equations are formed with various mathematical expressions, which can involve numbers, variables, and arithmetic operations like addition, subtraction, multiplication, or division.

• Left-hand side: Often contains variables and expressions that we manipulate.
• Right-hand side: Typically, this number or expression represents the conclusion or goal of the equation.

This particular equation says that when 23 is subtracted from our unknown number \(x\), the result is a specific number \(-8\). Understanding this balance is critical in algebra, as your goal is often to find the unknown, making both sides of the equation equal.

Solving Equations

Solving an equation means finding the value of the unknown that makes the equation true. Our goal in the given exercise is to determine what value \(x\) must be to satisfy the equation \(x - 23 = -8\). To do this, we employ the skill of isolating the variable. This involves getting \(x\) by itself on one side of the equation, making it easier to identify its value. You can think about it as a detective work.

• Step 1: Identify what needs to be moved to simplify the equation. In our example, we need to get rid of the \(-23\).
• Step 2: Perform inverse operations. To cancel \(-23\), we add 23 to both sides of the equation, maintaining the equation's balance.
• Step 3: Perform the operation: \(-8 + 23\), resulting in \(x = 15\).

This technique of balancing ensures that anything we do to one side of the equation must be done to the other side as well. Solving equations paves the way to decode more complex algebraic puzzles by expanding these basic concepts.

Introductory Algebra Concepts

Introductory algebra is where your journey in understanding mathematical relationships and their unknowns truly begins. It forms the bedrock upon which more advanced mathematical knowledge is built.

At its core, introductory algebra is about learning how to manipulate equations and understand variables and constants. Variables represent unknown values, while constants have fixed values. This fundamental aspect links directly to real-world scenarios where conditions are set, and we solve to find unknowns.

As you learn algebra:

• Practice translating words to math: Word problems turn into equations, just like how "a number minus 23 equals -8" becomes \(x - 23 = -8\).
• Isolate the variable: Develop skills to isolate the variable by adding, subtracting, multiplying, or dividing both sides as needed.
• Understand properties of equality: Know that the equation remains balanced as long as you perform the same operation to both sides.

Beginning with these foundational principles makes solving even intricate problems approachable, setting you up for success as mathematical topics increase in complexity.