Problem 8 - Algebra 1: Concepts and Skills

Question

Check to see if \(a=5\) is or is not a solution of the equation. $$ a+8=13 $$

Step-by-Step Solution

Answer

Yes, \(a=5\) is a solution to the equation \(a+8=13\).

Step 1: Substitute the given value

Substitute the given value \(a=5\) into the equation to check if it holds true. The modified equation becomes \(5 + 8 = 13\).

Step 2: Simplify LHS

Simplify the left-hand side (LHS) of the equation, that is \(5 + 8\). This adds up to 13.

Step 3: Compare LHS and RHS

Now compare the left-hand side (LHS) and the right-hand side (RHS) of the equation. If they are equal, then \(a=5\) is indeed the solution of the equation. In this case, LHS is 13 and RHS is also 13, hence, both are equal.

Key Concepts

Equations

Equations are mathematical statements that show the equality between two expressions. Most often, they involve one or more variables such as \(a\), \(x\), or \(y\). The equation given in the exercise is \(a + 8 = 13\). This is a simple linear equation because it involves a variable raised to the power of one.

Solving an equation means finding the value of the variable that makes the statement true. In our example, we are asked to find if \(a = 5\) is a solution. To determine a solution, we substitute the potential value into the equation. This is where techniques like the substitution method come into play.

Substitution Method

The substitution method is a technique used to solve equations by replacing the variable with a proposed value. It helps us check if a specific number can satisfy the equation. Let's break it down with our exercise as an example:

• **Substitute the Variable:** Begin by substituting the proposed value of the variable into the equation. In this case, we substitute \(a\) with \(5\), transforming our equation into \(5 + 8 = 13\).
• **Simplify the Expression:** Simplify the equation by performing the arithmetic operation. Adding \(5\) and \(8\) gives us \(13\).

Once we substitute and simplify, we are ready to move on to the verification of the solution. This ensures that the substituted value satisfies the equation.

Verification of Solutions

Verification of solutions is the process of confirming whether a substituted value actually solves the equation. It's the critical step that validates our result. Here's how you can do it for our equation example:

• **Compare Both Sides:** After substituting and simplifying, compare the left-hand side (LHS) of the equation to the right-hand side (RHS). If the two sides are equal, the proposed number is a solution. Here, both LHS and RHS simplify to \(13\), proving equality.
• **Confirm the Solution:** Since our LHS \(13\) equals RHS \(13\), it confirms that \(a = 5\) is indeed a solution of this equation.

Verification is important because it ensures that any arithmetic done during substitution was performed correctly. Without this step, you risk overlooking errors or false solutions.