Problem 23 - Algebra 1: Concepts and Skills

Question

Solve the equation. \(3 y+5=11\)

Step-by-Step Solution

Answer

The solution to the equation \(3y+5=11\) is \(y=2\).

Step 1: Isolate the y term

In order to find the value of the variable \(y\), first isolate the \(3y\) term by subtracting 5 from both sides of the equation: \(3y+5-5 = 11-5\). This simplifies to \(3y = 6\).

Step 2: Finalize the Solution

The next step is to isolate the \(y\) term by itself, which means getting rid of the 3 that is being multiplied with \(y\). We do that by dividing both sides of the equation with 3: \(\frac{3y}{3} = \frac{6}{3}\). This simplifies to \(y = 2\).

Key Concepts

Isolate the Variable

In algebra, solving an equation often revolves around isolating the variable. To isolate means to get the variable by itself on one side of the equation. This allows us to find out its specific value. When you have an equation like \(3y + 5 = 11\), your ultimate goal is to have \(y\) by itself.

Let's break this down in a simple, step-by-step way:

• First, identify the term that contains the variable. In our example, it's \(3y\).
• Next, eliminate any other numbers from the same side of the equation as the variable. Here, you need to subtract 5 from both sides, turning \(3y + 5 = 11\) into \(3y = 6\).
• The final stage in isolating the variable is to remove the coefficient – the number multiplying the variable. With \(3y = 6\), you divide both sides by 3, giving you \(y = 2\).

This process can be applied to any equation: identify, eliminate, and isolate. Remember, your operations must "balance" the equation; whatever you do to one side, you must do to the other.

Simplifying Expressions

Simplifying expressions is a crucial part of solving equations and one of the fundamental skills in algebra. By simplifying, you transform a complex expression into its simplest form. This makes it easier to work with.

To simplify, follow these key steps:

• Start by performing any arithmetic operations, like addition or subtraction, on both sides of the equation.
• Combine like terms to condense the expression. For instance, in \(3y + 5 - 5\), the +5 and -5 cancel each other out, leaving you with \(3y\).
• Continue simplifying until you can't simplify anymore and the expression is easy to understand and work with.

Understanding how to simplify expressions effectively can make solving any equation more straightforward and less intimidating, helping you to see the bigger picture and find solutions efficiently.

Basic Algebraic Operations

At the core of algebra are basic algebraic operations like addition, subtraction, multiplication, and division. These operations are the tools used to manipulate and solve equations. It’s important to master them because they're the foundation of everything you'll do in algebra.

Here's how each operation comes into play:

• Addition and Subtraction: Used to move terms from one side of the equation to the other. If you have a +5 on one side, you subtract 5 to eliminate it.
• Multiplication and Division: Used to get rid of coefficients of variables. In \(3y = 6\), divide by 3 to isolate \(y\).
• These operations need to be applied uniformly to maintain the equality of the equation.

Understanding and applying these basic operations is crucial in making your algebra journey smooth and successful. They help you isolate variables, simplify expressions, and ultimately, solve equations.