Problem 59 - 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 12 is subtracted from a number, the result is \(-2 .\) Find the number.

Step-by-Step Solution

Answer

The number is 10

Step 1: Formulate the Equation

Translate the statement 'If 12 is subtracted from a number, the result is -2' into a mathematical equation. Let \(x\) represents the unknown number, which can be formulated as \(x - 12 = -2\).

Step 2: Solve the Equation

To solve for \(x\), simplicity will be added on both sides of the equation, 12 is added on both sides to cancel them out from the left side. The resulting equation would be \(x = -2 + 12\).

Step 3: Simplify the Result

By calculating the right side of the equation, it results in \(x = 10\).

Key Concepts

Algebraic Expressions

Algebraic expressions are the backbone of algebra and are used to represent real-world problems in a mathematical format. They consist of variables, numbers, and operations (like addition and subtraction), but do not include an equality sign as equations do. They are crucial in forming equations that are then solved to find unknown values.

For example, the expression \( x - 12 \) in our exercise represents a number \( x \) reduced by 12 units. This expression by itself is not solvable because it is not equal to anything. However, once it becomes part of an equation, like \( x - 12 = -2 \), it is ready to be manipulated and solved. Understanding how to translate words into algebraic expressions and then into equations is essential for solving many algebraic problems.

Equation Solving Step by Step

Equation solving is a systematic process that involves finding the value of the unknown variable that satisfies the given equation. The step-by-step method is a simple yet effective way to approach solving linear equations, ensuring that every action taken moves towards isolating the variable and finding its value.

Starting with an equation like \( x - 12 = -2 \), the goal is to get \( x \) by itself on one side. To do this, one performs the same operation on both sides of the equation—here, adding 12 to each side to cancel out the -12. This step maintains the balance of the equation, which is crucial in algebra. After this, the equation simplifies to \( x = 10 \), revealing the value of the number. This approach, applied consistently, allows for successful solving of many types of algebraic equations.

Introductory Algebra

Introductory algebra serves as the foundation for all further study in mathematics. It introduces concepts of variables, equations, and functions. In the beginning stages, students learn about variables as placeholders for numbers and the importance of maintaining balance in equations. This fundamental understanding aids in expanding their problem-solving skills.

Through solving simple equations like \( x - 12 = -2 \) for \( x \) and finding that \( x = 10 \) when solved, students learn to interpret mathematical statements and to wield basic algebraic tools. Being comfortable with these introductory concepts is crucial as algebra grows in complexity with the addition of more variables, higher powers, and more intricate operations. Mastering the basics is the key to confidently navigating the more advanced territories of algebra.