Problem 9 - Algebra 1: Concepts and Skills

Question

Check to see if \(a=5\) is or is not a solution of the equation. $$ 27=36-2 a $$

Step-by-Step Solution

Answer

No, \(a=5\) is not a solution to the equation \(27=36-2a\).

Step 1: Identification and Substitution

Identify the given equation, which is \(27=36-2a\). We are given that \(a=5\). Substitute \(a=5\) into the equation, yielding \(27=36-2*5\).

Step 2: Simplification

Perform the calculation on the right side of the equation, \(36-2*5=26\). So, the equation becomes \(27=26\).

Step 3: Verification

The left hand side (LHS) and right hand side (RHS) of the equation do not match, because \(27 \neq 26\). Therefore, \(a=5\) is not a solution to the given equation.

Key Concepts

Understanding the Substitution Method

The Substitution Method is a fundamental technique for solving algebraic equations. The process begins by taking a known value and substiting it into the equation in place of the variable. In the context of our example, we're checking if a=5 is a solution for the equation 27 = 36 - 2a.

Here's how you implement the substitution method in this case:

• Begin by identifying the variable in the equation; here, it's a.
• Next, replace the variable with the given value; substitute a with 5.
• Now, the equation is personalized with the specific value, making it ready for simplification.

Substitution is particularly useful because it transforms abstract equations into concrete numerical expressions that can be calculated.

Simplifying Expressions

Simplifying expressions is the act of reducing an algebraic expression to its simplest form. This means performing all the arithmetic operations and combining like terms where possible. In our example, after substituting a with 5, we get the expression 36 - 2*5, which needs simplification.

To simplify, follow these steps:

• First, multiply 2 by 5 to get 10.
• Then, subtract this product from 36 to obtain 26.

The expression is now in its simplest form, making it easier to compare both sides of the equation. Simplification is crucial in algebra as it helps in identifying whether the equation holds true or not, as well as prepares the ground for further operations if needed.

Verifying Solutions

Verifying solutions is the final step in solving algebraic equations. It involves comparing the left and right sides of an equation to check for equality. If both sides match after simplification, the solution is correct; if not, it's incorrect. In the problem given, the simplified expression on the right yielded 26, which does not match the left side, which is 27.

It's important to note the following while verifying solutions:

• Ensure that all mathematical operations are carried out correctly during simplification.
• Compare the left and right sides of the equation meticulously.
• A mismatch indicates that the assumed value is not a solution.

Remember, verification is a powerful tool to confirm the accuracy of your results and should always be used to conclude the problem-solving process.