Problem 11 - Algebra 1: Concepts and Skills

Question

Check to see if \(a=5\) is or is not a solution of the equation. $$ 2 a+1=11 $$

Step-by-Step Solution

Answer

\(a=5\) is a solution of the equation \(2a+1=11\).

Step 1: Substituting

Substitute \(a=5\) into the equation \(2a+1=11\). Thus the equation becomes \(2*5+1=11\).

Step 2: Simplifying the Equation

On simplifying the equation, we get \(10+1=11\), which simplifies further to \(11=11\).

Step 3: Verifying the Solution

As both sides of the equation are equal, it holds. Hence, \(a=5\) is a solution of the given equation.

Key Concepts

Substitution Method

The substitution method is a key part of solving equations, especially when you want to find out whether a particular value, like \(a = 5\), satisfies an equation. Here's how to use it effectively:

• Identify the variable in the equation you need to replace. In our exercise, this is \(a\).
• Replace every occurrence of this variable in the equation with the proposed value. Substitute \(a = 5\) into the equation \(2a + 1 = 11\), giving us \(2 \times 5 + 1 = 11\).
• Ensure you've replaced the variable thoroughly and correctly to evaluate the equation without errors.

This method helps to see if the left-side expression equals the right side when a particular value is used, simplifying the process of verifying whether a value is a solution.

Simplifying Expressions

Once you've substituted the variable in the equation, simplifying the expression is the next key step. Simplification involves breaking down the equation into its simplest form to see if both sides are equal.

• Start with basic arithmetic operations, like multiplication and addition. Our equation, \(2 \times 5 + 1\), simplifies to \(10 + 1\).
• After simplifying further, you get \(11\), which is on the left side of the equation. The right side of the original equation is already 11.
• By simplifying, you make it easier to compare both sides of the equation to confirm their equality.

Being careful and systematic during simplification ensures accuracy and verifies if the substituted value works as a solution.

Verifying Solutions

The final step, verifying solutions, is crucial to confirm whether your substituted value is indeed a solution to the equation. Verification shows us that the entire process was executed correctly.

• After simplifying the expression, check if the left side equals the right side of the equation. In our example, \(11 = 11\), confirming that they are equal.
• If both sides are equal, like they are here, then the substituted value \(a = 5\) is a valid solution.
• If they don't match, then the value you proposed is not a solution, and you may need to try a different value or re-evaluate your steps.

This step is essential for ensuring your answer is correct and ready for presentation.