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When solving linear equations, identifying and combining like terms is a fundamental step. Like terms are terms that contain the same variables raised to the same power. For instance, in the equation \(8x - 2 = 7x - 5\), \(8x\) and \(7x\) are like terms because both contain the variable \(x\) to the power of one.

To combine them, you subtract \(7x\) from both sides, effectively isolating the variable on one side of the equation. This process simplifies the equation, making it easier to solve. You can consider it as condensing the equation by merging similar elements together. Combining like terms not only neatens the equation but also sets the stage for further operations needed to isolate the variable and find its value.

Here's why combining like terms is helpful:

• It reduces the complexity of the equation.
• It brings us one step closer to isolating the variable.
• It prepares the equation for further simplification.

Remember, only terms with the same variable and exponent can be combined!

After finding a potential solution for a linear equation, it's crucial to ensure its validity. Checking the equation is the process of verifying that your obtained solution indeed satisfies the original equation. This step is important as it confirms the accuracy of your algebraic manipulations.

To check the solution of an equation like \(8x - 2 = 7x - 5\), you would substitute the value of \(x\) into the original equation. If both sides of the equation balance after the substitution, your solution is correct. In our example, when we check \(x = -3\), it simplifies down to \(-26 = -26\), verifying that our solution is correct.

Benefits of checking your solution:

• Ensures that simple calculation errors are caught.
• Reinforces your understanding of the relationship between the variables and constants in an equation.
• Instills confidence in your problem-solving skills.

Always make a habit of checking your solutions to bolster your mathematical precision.

The goal of solving an algebraic equation is to find the value of the unknown variable that satisfies the equation. This process, known as finding algebraic solutions, involves a series of steps that typically begin with simplifying the equation and isolating the variable. In the given problem, we isolate \(x\) by combining like terms and then further simplify to find \(x = -3\).

The algebraic solution is the end result of those steps - the definitive value or set of values that satisfy the given equation. It's a representation of a quantity that was previously unknown and is now tangible and usable in various contexts. Once you have an algebraic solution, it can be used to solve related problems, or in real-world applications that the equation models.

Significance of finding algebraic solutions:

• Provides a concrete answer to abstract problems.
• It's fundamental in various applications, from basic mathematics to engineering and economics.
• Fosters a deeper understanding of mathematical relationships.

Finding algebraic solutions is not just about obtaining an answer; it's about developing the skills to systematically approach and solve problems.