Problem 29 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ x+5=15 $$

Step-by-Step Solution

Answer

The solution to the equation \(x + 5 = 15\) is \(x = 10\).

Step 1: Identify the equation

The equation provided is \(x+5 = 15\). Here, \(x\) is the unknown which needs to be found.

Step 2: Isolate the variable

Subtract 5 from both sides of the equation to isolate \(x\). This gives \(x = 15 - 5\).

Step 3: Solve for the variable

Subtracting 5 from 15 gives 10. Thus, \(x = 10\).

Key Concepts

Isolate the Variable

Isolating the variable is a fundamental step in solving linear equations. Essentially, it means to rearrange the equation so that the unknown variable—the one we're solving for—is by itself on one side of the equation. In our exercise, the equation is \(x + 5 = 15\). To isolate the \(x\), we need to move the 5 to the other side. How do we do that? By using the inverse operation, which, in this case, is subtraction. By subtracting 5 from both sides of the equation—because what we do to one side, we must do to the other—we are left with \(x = 10\). This clear, straightforward process highlights the variable, setting the stage for finding the solution.

Basic Algebra

Understanding the core principles of basic algebra is essential for approaching equations with confidence. At its heart, algebra involves using symbols, often letters, to represent numbers in equations, and performing operations with these symbols according to established mathematical rules.

When faced with our example, \(x + 5 = 15\), we see that 'x' is a placeholder for the unknown number we're trying to find. Basic operations such as addition, subtraction, multiplication, and division are used to manipulate these equations to find the value of 'x'. Our problem requires us to use addition and subtraction—the building blocks of algebra—to get our answer.

Equation Solving Step by Step

Solving an equation step by step prevents common mistakes and provides a strategy that can be applied to more complex problems. Each step should be performed with care and precision. In our example:

Step 1: Identify the equation

• Recognize that \(x + 5 = 15\) is a simple linear equation.

Step 2: Isolate the variable

• Determine that subtracting 5 from both sides will help isolate 'x'.

Step 3: Solve for the variable

• Understand that subtracting 5 from 15 gives us 10, thus \(x = 10\).

Following these steps helps cement a logical progression from problem to solution, instilling the skills needed to handle algebra with proficiency.

Subtraction Property of Equality

The subtraction property of equality holds that if you subtract the same amount from each side of an equation, the two sides remain equal. In other words, what you do to one side, you must do to the other to maintain balance. Consider the scales of a balance: if you remove the same weight from each side, the scales still balance.

In our example, we subtract 5 from both sides of the equation \(x + 5 = 15\) to maintain the equality, leading us to \(x = 10\). This principle is a cornerstone of algebra that ensures the integrity of the equation is preserved while we seek the unknown variable.