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The core idea of isolating variables in an equation is to get the unknown variable on one side by itself, surrounded by known numbers, or constants. Let's look at the example provided: \(5x = 20\). The goal is to find the value of \(x\), which requires isolating \(x\).

The first step to isolating \(x\) is to perform operations that help eliminate any number attached to \(x\), either through multiplication or division. In this case, \(x\) is being multiplied by 5. To isolate \(x\), we need to do the opposite operation, which is dividing by 5 on both sides. This keeps the equation balanced:

• Original Equation: \(5x = 20\)
• Divide both sides by 5: \(\frac{5x}{5} = \frac{20}{5}\)

Now, \(x\) is isolated on one side, ready for further solving. This simple trick of reversing operations is a staple in algebra.

The process of solving equations involves finding the value of the unknown variable that makes the equation true. Once we have isolated \(x\), we proceed to solve for its value.

In the equation \(\frac{5x}{5} = \frac{20}{5}\), we simply calculate the division to find \(x\). Doing the division on the right-hand side of the equation:\

• \(\frac{20}{5} = 4\)

Thus, we have solved the equation, and the value of \(x\) is 4. Solving equations often involves a series of operations, like addition, subtraction, multiplication, and division, applied to both sides of an equation. This ensures that the equality holds true throughout the process.

Simplification is a critical step in solving equations, and it involves making an equation as basic as possible. After isolating the variable and performing basic arithmetic operations, simplifying the equation confirms that the solution is easy to understand and apply.

In our given problem, after dividing both sides by 5, the equation \(\frac{5x}{5} = \frac{20}{5}\) simplifies to \(x = 4\). This indicates that \(x\) is by itself and has a clear, simplified numeric value of 4.

• Simplified Result: \(x = 4\)

Simplification helps in verifying your steps and ensures accuracy in complex problems. The simpler the equation, the easier it is to verify that all steps have been executed correctly.