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When solving linear equations, an important initial step is to combine like terms to simplify the expression. Like terms are terms that contain the same variable raised to the same power, though they might have different coefficients. For example, in the equation \(5x + 6x\), both terms have \(x\) as the variable.
To combine them, you add their coefficients together, so \(5x + 6x = 11x\). This makes the equation simpler and easier to solve. By ensuring all like terms are combined, you can clearly see the relationship between the variables and constants in your equation. Remember to do this for both sides of the equation separately, which helps in maintaining the balance of the equation.

After combining like terms, the next step is isolating the variable, which means getting the variable by itself on one side of the equation. This involves moving all terms without the variable to the opposite side of the equation. In our example, the equation was simplified to \(11x = -22\).
To isolate \(x\), we need to perform the reverse operation of what's currently affecting it. Since \(x\) is multiplied by 11, we divide both sides of the equation by 11 to solve for \(x\).

• This operation is key because it undoes the multiplication.
• Performing the same operation on both sides keeps the equation balanced.

By dividing, we eventually achieve \(x = -2\), which is the solution to the equation.

Solving linear equations involves a process of systematic steps, which ensure we find the correct solution. Here’s a brief rundown of the general steps:

• Identify and simplify: Rearrange the equation to identify like terms and simplify both sides separately. As in our example, this involves combining terms like \(5x\) and \(6x\) to get \(11x\).
• Rearrange and isolate: Move variables to one side and constants to the other. This usually involves adding or subtracting terms from both sides.
• Solve for the variable: Use inverse operations (like division or multiplication) to isolate the variable and find its value.

This step-by-step approach helps prevent errors and ensures consistency when approaching different equations. Understanding and practicing these steps build confidence and proficiency in solving all types of linear equations.