Problem 68 - Algebra 1: Concepts and Skills

Question

Use mental math to solve the equation. \(2 a=8\)

Step-by-Step Solution

Answer

The solution to the equation \(2 a = 8\) is \(a = 4\).

Step 1: Identify the equation

The equation provided is \(2 a=8\). We want to solve for 'a', meaning we want 'a' to be by itself on one side of the equation.

Step 2: Isolate the variable 'a'

To isolate 'a', we need to do the reverse operation of what is being done to 'a' in the equation. Since 'a' is being multiplied by 2, we should divide by 2. It's vital to remember that whatever we do to one side of the equation, we must also do to the other side to keep it balanced. So, if we divide the left side of the equation by 2, we must also divide the right side by 2.

Step 3: Solve for 'a'

By following the operation in step 2, we end up with \((2a) / 2 = 8 / 2\), which simplifies to \(a = 4\).

Key Concepts

Isolating the Variable

When solving algebraic equations, a key step is often to isolate the variable. This means rearranging the equation so that the variable we're solving for is by itself on one side. Take for instance when we're faced with the problem 2a = 8. Our goal is to isolate a. How do we achieve that? It's simple: think of the equation as a set of balanced scales. Whatever operation is applied to the variable, we want to perform the reverse, to 'free' it.

As illustrated in our example, successfully isolating the variable requires executing an operation that will counteract the one applied to the variable. Since a is multiplied by 2, we must divide by 2 to counter this. Remember, the endgame is to have a standing alone, which signifies that it has been isolated.

Balance the Equation

The concept of balancing the equation is foundational in algebra. Imagine there is an actual scale with weights equally distributed on both sides; that's our equation. Removing or adding weight from one side would tip the scale—and it's the same with equations. To maintain balance, any operation we perform on one side must also be mirrored on the other.

For the operation of dividing by 2 that we used to isolate 'a' in our equation 2a = 8, we must divide both sides by 2 to keep things fair and square. Balancing ensures that the equation remains true throughout our manipulation, leading to a correct solution. It's about maintaining equality; after all, that's the heart of what an equation represents.

Reverse Operations in Algebra

To solve equations, we often use the strategy of reverse operations. This involves doing the opposite of what's currently affecting the variable. If it's addition, we subtract. If it's multiplication, we divide, and so on. In our equation 2a = 8, the variable a is being multiplied by 2, so we perform the reverse—division—to nullify that multiplication.

This act of reversing can be thought of as peeling layers off an object to get to the core; we are essentially peeling away the mathematical operations to reveal the value of the variable. Understanding reverse operations helps us navigate through more complex equations where multiple, successive operations need to be reversed. It's a critical skill in the algebraist's toolkit, allowing for a systematic approach to find solutions.