91Ó°ÊÓ

Solving linear equations is all about finding the value of the variable that makes the equation true. A linear equation looks like the equation provided in the exercise: \(9x + 1 = 8x + 4\). Linear equations have variables that are only to the power of 1. The main goal is to manipulate the equation in such a way that we can isolate the variable on one side, making it easy to find its value.

When tackling any equation, follow these general steps:

• Combine like terms
• Use basic arithmetic operations
• Isolate the variable

By consistently applying these steps, solving linear equations becomes a straightforward process.

Isolating the variable is a crucial step in equation solving. This means getting the variable alone on one side of the equation so that you can determine its value. Here's how it works using the provided exercise:

First, let's move the terms involving the variable to one side. We had: \(9x + 1 = 8x + 4\). Subtract \(8x\) from both sides to simplify: \(9x + 1 - 8x = 8x + 4 - 8x\). This leaves us with: \(x + 1 = 4\).

Next, we need to get \(x\) by itself. To do this, subtract 1 from both sides: \(x + 1 - 1 = 4 - 1\). Simplifying this gives us: \(x = 3\). Now, we've isolated the variable and found that \(x = 3\). This balance is key in mathematics. By doing the same operation on both sides, we ensure the equality stays true.

Once you've found the value of the variable, it's important to verify it. Verifying your solution ensures that your calculations are correct and the solution works for the original equation.

We'll substitute \(x = 3\) back into the original equation: \(9x + 1 = 8x + 4\). Plugging in the value: \(9(3) + 1 = 8(3) + 4\). Simplify both sides: \(27 + 1 = 24 + 4\). This further simplifies to: \(28 = 28\).

Since both sides are equal, our solution \(x = 3\) is verified to be correct. Always go through this last step to confirm your solution. It helps catch errors and reinforces your understanding of the problem.