Problem 120 - Introductory and Intermediate Algebra for College Students 4th

Question

In each exercise, determine whether the given number is a solution of the equation. $$-\frac{1}{2}=x-\frac{2}{3} ; \frac{1}{6}$$

Step-by-Step Solution

Answer

\(\frac{1}{6}\) is a solution of the equation \(-\frac{1}{2}=x-\frac{2}{3}\)

Step 1: Substitute the given value

In the equation \(-\frac{1}{2}=x-\frac{2}{3}\), substitute \(x\) with the given number \(\frac{1}{6}\). So, the equation becomes \(-\frac{1}{2}=\frac{1}{6}-\frac{2}{3}\)

Step 2: Calculating the values

Now, simplify the right side of the equation to see what it equals. \(\frac{1}{6}-\frac{2}{3}=-\frac{1}{2}\)

Step 3: Comparing the values

Finally compare the obtained solution with the left side of the equation. If they are equal, this means the given number is a solution. In this case, \(-\frac{1}{2}=-\frac{1}{2}\)

Key Concepts

Substitution Method

The substitution method is a powerful tool used in solving equations where you replace variables with specific values to test solutions. This method is straightforward. You take a number that you suspect might be a solution to the equation and replace the variable with that number.
Let's consider our equation: \(-\frac{1}{2} = x - \frac{2}{3}\). The task is to check if \(\frac{1}{6}\) is a solution.
We proceed by substituting \(x\) with \(\frac{1}{6}\) to get: \(-\frac{1}{2} = \frac{1}{6} - \frac{2}{3}\).
This substitution results in a new equation entirely involving numbers on both sides, making it possible to verify whether the equality holds true. It's like testing whether a key fits in a lock.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions to their simplest form. This usually means having the same denominator before you subtract or add fractions.
In our example, after substitution, the right side of the equation becomes \(\frac{1}{6} - \frac{2}{3}\). To simplify this:

• First, determine a common denominator for both fractions, which in this case is \(6\).
• Adjust \(\frac{2}{3}\) to have this common denominator: \(\frac{2}{3} = \frac{4}{6}\).

Now, you subtract: \(\frac{1}{6} - \frac{4}{6} = -\frac{3}{6}\).
Further, \(-\frac{3}{6}\) simplifies to \(-\frac{1}{2}\), matching the left side of the equation. Proper simplification is crucial, as it confirms whether both sides of the equation are indeed equal.

Solution Verification

Solution verification is the final step where you check if your solution holds true in the original equation. This ensures the number tested truly satisfies the equation.
In this exercise, once the right side of the equation \(\frac{1}{6} - \frac{2}{3}\) simplifies to \(-\frac{1}{2}\), we compare it to the left side, which is already \(-\frac{1}{2}\).
If both sides are equal as they are here, it confirms that the number tested, \(\frac{1}{6}\), is indeed a solution for the equation. Verification reassures us that the substitution and simplification steps were done correctly and the solution is valid.