Problem 13 - Algebra 1: Concepts and Skills

Question

Check to see if \(a=5\) is or is not a solution of the equation. $$ 5 a+4=26 $$

Step-by-Step Solution

Answer

No, \(a=5\) is not a solution of the equation \(5a+4=26\).

Step 1: Substituting the value

To test whether \(a=5\) is a solution, substitute \(a=5\) in the equation \(5a+4=26\). So, the equation becomes \(5(5)+4=26\).

Step 2: Simplifying the equation

Simplify the equation obtained in step 1. This gives \(25+4=26\). Simplifying this gives \(29=26\).

Step 3: Validating the solution

Since \(29\) is not equal to \(26\), \(a=5\) is not a solution of the equation \(5a+4=26\).

Key Concepts

Substitution Method

The substitution method is a straightforward technique often used to determine if a proposed value is a solution to an equation. To apply this method, you simply replace the variable in the equation with the given value and check if the resulting expression is true. In our exercise, we substituted the value of \(a=5\) in the equation \(5a+4=26\) by replacing the \(a\) with 5, which resulted in the equation \(5(5)+4=26\). This step is crucial because it allows you to directly assess whether your proposed solution satisfies the equation.

By performing a substitution, you're essentially testing whether the provided value will make the left side of the equation equal to the right side, ensuring an accurate solution. It's important to perform this step carefully as substitution errors can lead to incorrect conclusions.

Equivalent Equations

When dealing with equations, it's helpful to understand the concept of equivalent equations. Equivalent equations are different equations that have the same solution or solutions. They arise frequently in algebra when manipulating equations to simplify or solve them.

For example, if we manipulate \(5a + 4 = 26\) by subtracting 4 from both sides, we obtain \(5a = 22\). Both are equivalent because they will result in the same solutions for \(a\). However, in our original exercise, we found that substituting \(a=5\) into the first form does not satisfy the equation.

The process of finding equivalent forms of equations can simplify the problem and make checking potential solutions more manageable. It helps to see if other forms of the same equation yield consistent solutions or validate an initial test.

Checking Solutions

Checking a solution is a verification step in solving equations. This step helps confirm whether the proposed value for the variable actually satisfies the equation. In our exercise, after substituting \(a=5\), we simplified the expression to \(29=26\). Clearly, 29 is not equal to 26, indicating \(a=5\) is not a solution.

This process of checking is about validating the equation's balance or equality. If both sides of the equation are equivalent after substitution, then the tested value is indeed a solution. If they are not, as they weren't in this example, the value is not a solution.

Always remember to substitute back into the original equation for a final check, especially after transformations, to ensure no mistakes have been made. This step bolsters confidence in your solution process and final answer.