Problem 18 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$8 x-16=0$$

Step-by-Step Solution

Answer

The solution is x = 2.

Step 1: Rearrange the Equation

Begin by adding 16 to both sides of the equation to isolate the term with x. This yields the new equation \(8x = 16\).

Step 2: Solve For x

Next, divide both sides of the equation by 8 to solve for x, resulting in \(x = 2\).

Step 3: Check the Solution

Substitute x = 2 back into the original equation \(8x - 16 = 0\), to verify the solution. Doing so yields \(8(2) - 16 = 0\), which simplifies to \(16 - 16 = 0\), confirming that x = 2 is the correct solution.

Key Concepts

Checking Solutions

When solving linear equations, it is very important to check your solutions. This ensures that the solution is correct, and it fits the original equation. Checking your solution is a relatively simple but crucial step. Here’s how:

First, substitute the solution back into the original equation. This means taking the value you found for the variable and putting it back in the equation in place of the variable. Then, perform the calculations and see if both sides of the equation balance or equal the same number. If they do, your solution is correct.

For example, in the equation \(8x - 16 = 0\), after solving, we found \(x = 2\). By substituting \(x = 2\) back into the original equation, you will have \(8(2) - 16 = 0\). Solve the left side of the equation to get \(16 - 16 = 0\). Since both sides equal zero, \(x = 2\) is indeed the correct solution. Checking is like a tag-along friend ensuring you're on the right track.

Isolation of Variables

Isolation of variables is a fundamental technique used in solving linear equations. The main goal is to get the variable of interest on one side of the equation, “alone.”

Start by identifying the operation on the variable that needs to be reversed. For example, in the equation \(8x - 16 = 0\), the aim is to solve for \(x\). The term \(-16\) is subtracted from \(8x\). To reverse this, you should add 16 to both sides of the equation. Doing this gives you \(8x = 16\).

By isolating the variable, you’re simplifying the equation and preparing it for the next step, which is solving for the exact value. The idea is to "peel away" all other numbers and operations around the variable, like layers of an onion.

Algebraic Manipulation

Algebraic manipulation refers to the techniques used to rearrange and simplify equations in order to solve them. This involves performing various operations to both sides of the equation to keep them balanced.

In our example, after isolating the term with \(x\), the equation becomes \(8x = 16\). The next step is to solve for \(x\) by dividing both sides by 8. This is an example of algebraic manipulation, which yields \(x = 2\).

Key techniques in algebraic manipulation include:

• Adding or subtracting the same number from both sides of the equation.
• Multiplying or dividing both sides by the same number to maintain balance.

These techniques allow you to maintain the equality by keeping both sides of the equation equivalent through each manipulation. Understanding these basic operations is essential for solving more complex equations as well.