Problem 29 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$-\frac{1}{5}+y=-\frac{3}{4}$$

Step-by-Step Solution

Answer

The solution to the equation \(-\frac{1}{5}+y=-\frac{3}{4}\) is \(y=-\frac{11}{20}\).

Step 1: Identify Equation

The equation in question here is \(-\frac{1}{5}+y=-\frac{3}{4}\). The goal is to solve for \(y\).

Step 2: Adding \(\frac{1}{5}\) to Both Sides

In order to isolate \(y\), add \(\frac{1}{5}\) to both sides of the equation by using the addition property of equality. This gives us \(y=-\frac{3}{4}+\frac{1}{5}\).

Step 3: Simplifying

Simplify the equation by adding the fractions on the right side. In order to add two fractions, they must have the same denominator. So convert \(-\frac{3}{4}\) and \(\frac{1}{5}\) to have a common denominator, like 20 in this case, to get \(y=-\frac{15}{20}+\frac{4}{20}=-\frac{11}{20}\).

Step 4: Checking the Solution

To ensure that the solution is correct, substitute \(-\frac{11}{20}\) for \(y\) into the original equation. Doing so results in \(-\frac{1}{5}-\frac{11}{20}=-\frac{3}{4}\), which is a true statement. So, the solution of the equation is indeed \(y=-\frac{11}{20}\).

Key Concepts

Equation Solving

When solving equations, the primary goal is to isolate the variable. Here, we want to solve the equation \(-\frac{1}{5} + y = -\frac{3}{4}\) for \(y\). To achieve this, we apply the **Addition Property of Equality**, which allows us to add the same quantity to both sides of the equation without altering its balance. This method is essential because it maintains the equality of the equation. For our specific equation:

• We add \(\frac{1}{5}\) to both sides of the equation to get \(y = -\frac{3}{4} + \frac{1}{5}\).
• This step helps in getting \(y\) all by itself on one side of the equation.

By isolating \(y\), we are one step closer to finding its value.

Fraction Operations

Working with fractions requires a few additional steps, especially when you need to add or subtract them. In the equation we are tackling, \(y = -\frac{3}{4} + \frac{1}{5}\), fractions are involved. To simplify this:

• First, identify a common denominator for the fractions \(-\frac{3}{4}\) and \(\frac{1}{5}\). In our case, the least common denominator is 20.
• Convert each fraction to equivalent fractions with this common denominator: \(-\frac{3}{4}\) becomes \(-\frac{15}{20}\) and \(\frac{1}{5}\) becomes \(\frac{4}{20}\).
• Now, add these two fractions: \(-\frac{15}{20} + \frac{4}{20} = -\frac{11}{20}\).

This operation results in \(y = -\frac{11}{20}\). Understanding fraction operations is key to solving such equations correctly.

Checking Solutions

Once we have a proposed solution, it is crucial to verify its accuracy. Substituting our found value for \(y\), which is \(-\frac{11}{20}\), back into the original equation \(-\frac{1}{5} + y = -\frac{3}{4}\), serves this purpose.

• Substitute \(-\frac{11}{20}\) for \(y\): \(-\frac{1}{5} - \frac{11}{20} = -\frac{3}{4}\).
• To confirm, convert \(-\frac{1}{5}\) to a fraction with the denominator 20, which is \(-\frac{4}{20}\).
• Add \(-\frac{4}{20}\) and \(-\frac{11}{20}\) to get \(-\frac{15}{20}\), simplifying this gives \(-\frac{3}{4}\).
• The original equation is satisfied, confirming that our solution is correct.

Checking solutions ensures that we haven't made errors in our calculations, giving us confidence in our result.