Problem 45 - Algebra 1: Concepts and Skills

Question

Check to see if the given value of the variable is or is not a solution of the equation. \(\frac{r^{2}}{2}=40 ; r=9\)

Step-by-Step Solution

Answer

No, \( r = 9 \) is not a solution.

Step 1 Substitute

Substitute the given value, \( r=9 \), into the equation \( \frac{r^{2}}{2}=40 \). Therefore, the equation becomes \( \frac{9^{2}}{2}=40 \).

Step 2 Simplify

Now simplify the left hand side of the equation. This gives \( \frac{81}{2}=40.5 \).

Step 3 Compare

Compare the calculated value with the right-hand side of the equation. As \( 40.5 \neq 40 \), the given value \( r=9 \) is not a solution to the equation.

Key Concepts

Substitution Method

The substitution method is an essential technique in algebra, particularly when it comes to solving equations. It involves replacing variables with numbers or other expressions to simplify a problem. This method can be especially useful when dealing with quadratic equations or systems of equations. To apply it correctly, it's important to follow every step with precision.

In our example, the substitution method is used to check if a given value is a solution to the equation. By substituting the variable, in this case, substituting nine for the variable r, we transform the abstract equation into a concrete arithmetic problem. This is a straightforward application, but substitution can also be used in more complex scenarios such as when one equation in a system is solved for one variable and that variable expression is substituted into the other equation.

Simplifying Expressions

Simplifying expressions is the process of reducing complexity in an algebraic expression, making it easier to understand or solve. Simplification may involve combining like terms, using exponent rules, or carrying out arithmetic operations. The main goal is to make the expression as compact and straightforward as possible, without changing its value.

In our exercise, simplification comes down to arithmetic since we are working with numerical values after applying the substitution method. We calculate the square of nine to get 81 and then divide by two to arrive at 40.5. This step is vital as it brings the equation to a form where we can directly compare both sides to verify if the initial value is indeed a solution. A common mistake during simplification is to rush and make arithmetic errors, so it’s key to be methodical and double-check each operation.

Verifying Solutions

Verifying solutions is the final check in the problem-solving process where we confirm whether our answer logically satisfies the original equation. This involves a comparison between what the equation should result in, with what we have after applying our proposed solution.

In this case, we compare the simplified expression on the left side, which is 40.5, to the constant on the right side, which is 40. As they do not match, we conclude that the proposed solution, r=9, does not satisfy the equation. Verifying is not a mere formality; it is crucial. It helps catch potential mistakes made while solving the equation and gives confidence in the final answer. If a proposed solution is incorrect, as in this example, we must revisit previous steps or try different solutions.