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Algebraic expressions are combinations of variables, numbers, and operations (like addition, subtraction, multiplication, and division). They can represent real-life scenarios, or simply provide a formula for a mathematical relationship. In the given exercise, -2(a^{2}+3a-1)+2a^{2}+7a is an algebraic expression. Understanding how to manipulate these expressions is crucial for solving algebra problems.

An important element to note is the use of parentheses, which indicates the terms inside should be dealt with first before applying any operation outside the parentheses, as per the order of operations (PEMDAS/BODMAS). Distributing a factor across terms within parentheses is a common starting point for simplifying algebraic expressions, as seen in Step 1 of the solution.

Combining like terms is a method of simplifying algebraic expressions, key to solving equations efficiently. Like terms are terms that have the same variables raised to the same power. They can be combined because they represent the same quantities. For example, 2x and 3x are like terms and can be added together to get 5x, but 2x^2 and 3x are not like terms and cannot be combined.

Why Combine Like Terms?

• It simplifies the expression.
• It makes the following steps in solving the equations clearer and easier.
• It is necessary to isolate the variable when solving for it.

In the problem provided, Step 2 illustrates combining -2a^2 with 2a^2 and -6a with 7a to simplify the expression into a linear equation.

Linear equations are algebraic equations where each term is either a constant or the product of a constant and a single variable. A linear equation looks like ax + b = 0, where a and b are constants. The solutions to these types of equations are the basis of linear algebra and can represent lines on a graph.

Solving linear equations typically involves isolating the variable on one side of the equation. In the exercise's Step 3, after combining like terms, we obtained a + 2 = 0, a simple linear equation. Subtracting 2 from both sides gives us the solution a = -2. A good practice is always to check the solution as done in Step 4, to ensure that it satisfies the original equation.