Problem 15 - Algebra 1: Concepts and Skills

Question

Check to see if \(a=5\) is or is not a solution of the equation. $$ a^{2}+2=27 $$

Step-by-Step Solution

Answer

Yes, \(a = 5\) is a solution to the given equation.

Step 1: Substitute the value

Substitute \(a=5\) into the given equation, \(a^{2}+2=27.\) It becomes \((5)^{2}+2=27, which simplifies to 25+2=27.

Step 2: Simplify the left-hand side

Simplify the left-hand side to get \(25 + 2 = 27\).

Step 3: Check if left side equals right side

Check if the left side of the equation is equal to the right side. Here, the left side simplifies to 27, which indeed equals to the right side of the equation.

Key Concepts

Substitution Method

The substitution method is a fundamental technique used in algebra to solve equations, especially useful when dealing with quadratic equations. It involves replacing a variable with its known or assumed value to determine whether the solution is valid or not.

For instance, given an equation like \(a^2 + 2 = 27\), the first step using substitution is to take the value we want to check, in this case, \(a=5\), and replace the variable \(a\) with 5 in the equation. This step transforms the abstract equation into a concrete arithmetic problem that we can easily solve. It's akin to taking a general key and custom-making it to fit a specific lock. Here, you transform \(5^2 + 2\) into \(25 + 2\), simplifying the process of finding the solution.

Simplifying Expressions

Simplifying expressions is like tidying up a room: it involves combining like terms and reducing expressions to their simplest form to make them easier to understand and work with. In the context of solving equations, simplification helps us see the core components of the expression, thus making it clearer whether or not we have the correct solution.

Take the expression \(25 + 2\); it looks simple already, but the principle remains important. When expressions become more complex, combining like terms and reducing the expression can be crucial in solving for the variable. For example, if we had \(25a + 2a\), we would combine like terms to get \(27a\), consolidating our expression and bringing us a step closer to finding the value of \(a\). It's essentially decluttering the mathematical 'room' to make our final destination more accessible.

Validating Solutions

Validating solutions is the final checkpoint in the journey of solving an equation. It's like checking your answers in a quiz to make sure they are correct before handing it in. After you've done the work of solving an equation, you must plug your solution back into the original equation to see if it makes both sides of the equation equal.

In our example, after substitution and simplification, we get \(27 = 27\). This equality tells us that the solution \(a = 5\) is indeed correct. However, if we ended up with \(27 eq 27\), that would mean our solution is incorrect. Just like proofreading an essay, validating our solution ensures that our final answer is accurate and makes sense within the confines of the equation we're working with.