Problem 101 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(97-108,\) determine whether the given number is a solution of the equation. $$5(w+3)=2 w-21 ;-10$$

Step-by-Step Solution

Answer

-10 is not a solution for the given equation.

Step 1: Substitute the Given Value

Start by substituting the given value, -10, into the equation. This gives us \(5(-10+3) = 2(-10) - 21\).

Step 2: Simplify the Equation

Simplify both sides of the equation. We get: \(5*(-7) = -20 - 21\) which simplifies further to \(-35 = -41\).

Step 3: Validate the Result

Compare if the left-hand side of the equation is equal to the right-hand side. In this case, we can see that -35 is not equal to -41.

Key Concepts

Substitution Method

When working with algebraic equations, the substitution method is a pivotal technique for finding whether a particular value is a solution to the equation. The process involves taking the given value and replacing the variable in the equation with this value. For instance, if you have an equation like 5(w+3)=2w-21 and you want to check if -10 is a solution, you would replace every occurrence of w with -10.

The benefit of this method is that it transforms the abstract equation into a concrete arithmetic problem, letting us compute and compare numbers directly. Remember, the goal is to achieve a true statement: both sides of the equation should yield the same number after the substitution and simplification are complete. If they do not, as with -35 ≠ -41, we know that the value is not a solution to the equation. This method is not only used for validating solutions of an equation but also in solving systems of equations, where one variable can be expressed in terms of another and substituted into a different equation.

Simplifying Equations

Simplifying equations is another core concept in algebra that aids in understanding and solving equations more efficiently. It involves reducing an equation to its simplest form by performing arithmetic operations and combining like terms. For the equation given in the exercise, 5(w+3)=2w-21, simplifying is done by multiplying out parentheses and consolidating terms on both sides of the equation.

After substituting w with -10, we get 5(-10+3) on the left and 2(-10)-21 on the right. We must multiply the numbers inside the parenthesis by the factor outside, and add or subtract the resultant products accordingly. The aim is to have the equation in a form where obvious arithmetic results in a clear, numeric comparison. By breaking down the problem into simpler parts, students can avoid common errors and more easily verify whether their final simplified result makes sense.

Validating Solutions

Finally, validating solutions is the process used to check the accuracy of the found solution by ensuring it satisfies the original equation. After substituting and simplifying, if both sides of the equation equal the same number, the value is a valid solution. To illustrate using the given exercise, after simplification, we obtain -35 on the left and -41 on the right. These are not equal; hence, we conclude that -10 is not a solution to the equation.

Validation is a critical final step because it confirms that no mistakes were made during the substitution and simplification processes. It builds confidence in the students’ algebraic manipulations and, when the solutions are valid, reaffirms their understanding of the concepts. It's a practice that underscores the importance of careful and methodical problem-solving in algebra.