Problem 70 - 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. \(17-3 w=2 ; w=5\)

Step-by-Step Solution

Answer

Yes, the given value \(w = 5\) is a solution of the equation \(17 - 3w = 2\).

Step 1: Substitute the given value into the equation

Replace the variable 'w' in the equation \(17 - 3w = 2\) with the given value, which is 5. The equation becomes \(17 - 3*5 = 2\), which simplifies to \(17 - 15 = 2\).

Step 2: Simplify and check

Simplify the left side of the equation. We get \(2 = 2\). Since both sides of the equation are equal, we can conclude that the given value of 'w' is indeed a solution to the equation.

Key Concepts

Substitution Method

One of the fundamental techniques in solving algebraic equations is the substitution method. This method involves replacing variables with their given values to see if the equation holds true. It's like swapping out a puzzle piece to see if it fits. For instance, when you have an equation like \(17 - 3w = 2\), and you're given a value for \(w\), such as \(w=5\), you substitute 5 in place of \(w\) to determine if it satisfies the equation.

This method is particularly useful because it allows you to work with simpler, numerical expressions, making it easier to see if your solutions are correct. It is frequently the first step in the process of evaluating equations and is essential for verifying possible solutions.

Simplifying Expressions

After substituting the values into the equation, the next vital step is simplifying expressions. Simplifying means performing all the arithmetic operations in the expression to reduce it to its most basic form. In the process of simplification, you apply basic arithmetic rules – addition, subtraction, multiplication, and division – to combine like terms and eliminate parentheses.

Consider our example where, after substitution, we get \(17 - 3*5\). We simplify the expression by multiplying 3 by 5 to get 15, and then subtract 15 from 17 to get 2. Simplification makes it clear whether both sides of the equation balance or not, hence confirming the validity of the initial substitution.

Verifying Solutions

The last step, verifying solutions, is about checking whether your answer makes the original equation true. In other words, after the substitution and simplification steps, you need to compare the simplified version of the original equation to see if the left side is indeed equal to the right side.

In our example, the equation \(17 - 3w = 2\) becomes \(2 = 2\) after the value \(w = 5\) is substituted and the expression is simplified. Because both sides are equal, we confirm that \(w = 5\) is a solution to the equation. Verification is a proof that your calculations are correct, and in a broader sense, it reassures confidence in the process of solving algebraic equations.