Problem 30 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation. Using the addition property of equality. Be sure to check your proposed solutions. $$-\frac{1}{8}+y=-\frac{1}{4}$$

Step-by-Step Solution

Answer

The solution to this equation is \( y= -\frac{1}{8} \)

Step 1: Apply the Addition Property of Equality

Add \(\frac{1}{8}\) to both sides of the equation to isolate 'y' on one side. The equation becomes: \(y=-\frac{1}{4}+\frac{1}{8}\).

Step 2: Simplify the Right Side of the Equation

Now, proceed to add the fractions on the right side of the equation, to get the value of 'y'. Remember to find a common denominator to add fractions. Since the denominators are 4 and 8, the common denominator will be 8. The simplified form of the right side is \(y=-\frac{2}{8}+\frac{1}{8}\).

Step 3: Calculate 'y'

Continue with the fraction calculation: \(y=-\frac{2}{8}+\frac{1}{8} = -\frac{1}{8}\).

Step 4: Verify the Solution

Plug the solution \(y=-\frac{1}{8}\) back into the original equation in place of 'y' and check if both sides of the equation have the same value. The equation `-1/8 + (-1/8) = -1/4` verifies that the solution is correct as both sides are equal.

Key Concepts

Addition Property of Equality

The Addition Property of Equality is a fundamental principle used to solve algebraic equations. The idea is straightforward: whatever you add to one side of an equation, you must also add to the other side to maintain balance. Think of an equation as a set of scales; to keep them level, whatever weight you add on one side has to be added to the other. This property allows us to manipulate equations to isolate the variable we need to solve for.

For example, in the given exercise \( -\frac{1}{8} + y = -\frac{1}{4} \) we're looking to find the value of 'y'. According to the Addition Property of Equality, we can add \( \frac{1}{8} \) to both sides of this equation without changing the equality, setting the stage to solve for 'y'. Let's apply this property step by step:

• Add \( \frac{1}{8} \) to each side of the equation.
• This results in \( y = -\frac{1}{4} + \frac{1}{8} \).

Once we've isolated 'y', we can move on to simplifying the right side to solve the equation completely.

Simplifying Fractions

Simplifying fractions is a key skill in algebra that makes equations easier to work with. It involves reducing fractions to their simplest form and is especially handy when dealing with the addition or subtraction of fractions. The goal is to have the smallest numerator and denominator possible without changing the value of the fraction.

In our exercise, to simplify \( -\frac{1}{4} + \frac{1}{8} \), we need a common denominator to combine these two fractions. Here are the steps we can follow:

Finding a Common Denominator

Identify the least common multiple of the denominators. In this case, the denominators are 4 and 8, and the least common denominator is 8.

Adjusting Fractions

Rewrite the fractions so they both have the denominator of 8: \( -\frac{2}{8} \) and \( \frac{1}{8} \).

Combining Fractions

Now, we can simply add the numerators: \( -2 + 1 = -1 \), leading to the simplified fraction \( -\frac{1}{8} \).
Simple fractions are much easier to work with and lead to clearer, more straightforward solutions.

Verifying Algebraic Solutions

Once you believe you have solved an algebraic equation, it is paramount to verify the solution to ensure it is correct. Verifying algebraic solutions involves substituting the variable's value back into the original equation to check if the equation holds true. It's the ultimate test to confirm whether the solution process was carried out accurately.

In our exercise, after finding that \( y = -\frac{1}{8} \), we verify it by substituting this value back into the original equation:

• Replacing 'y' with \( -\frac{1}{8} \), the original equation becomes \( -\frac{1}{8} + -\frac{1}{8} \).
• Simplify the left side to \( -\frac{2}{8} \), which reduces to \( -\frac{1}{4} \).
• If the left side equals the right side of the original equation \( -\frac{1}{4} \), then our solution is correct.

Through this process, we verify that \( y = -\frac{1}{8} \) solves the equation, ensuring our algebraic work holds up to scrutiny.