Problem 43 - Algebra A Combined Function

Question

Solve each equation. See Examples 9 and \(10 .\) \(\frac{1}{2}(2 x-1)=-\frac{1}{7}-\frac{3}{7}\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{-1}{14}\).

Step 1: Simplify both sides of the equation

Start by simplifying the right side of the equation. Combine \(\frac{-1}{7}\) and \(\frac{-3}{7}\) on the right side:\[\frac{-1}{7} - \frac{-3}{7} = \frac{-4}{7}\]

Step 2: Expand the equation

Expand the left side of the equation using the distributive property:\[\frac{1}{2}(2x-1) = 1x - \frac{1}{2}\]Thus, the equation becomes:\[1x - \frac{1}{2} = \frac{-4}{7}\]

Step 3: Eliminate the fraction from the left side

To eliminate the fraction on the left, add \(\frac{1}{2}\) to both sides of the equation:\[1x = \frac{-4}{7} + \frac{1}{2}\]

Step 4: Find a common denominator and simplify

To simplify \(\frac{-4}{7} + \frac{1}{2}\), find a common denominator (14 in this case):\[\frac{-4}{7} = \frac{-8}{14}\, \quad \frac{1}{2} = \frac{7}{14}\]Add these:\[\frac{-8}{14} + \frac{7}{14} = \frac{-1}{14}\]

Step 5: Solve for x

Now the equation is:\[x = \frac{-1}{14}\]This gives the final solution for x.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra used to simplify expressions. It states that multiplying a sum by a number gives the same result as multiplying each addend individually by the number and then adding the products. This property is represented by the formula: \( a(b + c) = ab + ac \).
In our exercise, the distributive property is applied to \( \frac{1}{2}(2x - 1) \). This involves multiplying each term inside the parentheses by \( \frac{1}{2} \):

• \( \frac{1}{2} \times 2x = 1x \)
• \( \frac{1}{2} \times -1 = -\frac{1}{2} \)

This allows us to rewrite the expression without parentheses, making it easier to solve the equation. The distributive property is a powerful tool in algebra for simplifying complex expressions and equations.

Simplifying Fractions

Simplifying fractions involves combining fractions to create a simpler form. It is essential to simplify fractions to make mathematical expressions easier to work with.
In the exercise, we encounter fractions on both sides of the equation. On the right side, the fractions \( \frac{-1}{7} \) and \( \frac{-3}{7} \) are combined. Since they have the same denominator, simplification is straightforward:

• Add the numerators: \(-1 + (-3) = -4 \)
• Keep the denominator: \(7\)
• The result is \(\frac{-4}{7}\)

It's important to remember that when combining fractions, the denominators must be the same. This is a basic rule for performing operations with fractions.

Common Denominator

Finding a common denominator is crucial when adding or subtracting fractions that do not share the same denominator. A common denominator allows you to rewrite fractions so they are comparable and can be combined.
In our example, we need to add \( \frac{-4}{7} \) and \( \frac{1}{2} \). The denominators are different, so we find a common denominator by determining the least common multiple (LCM) of 7 and 2, which is 14. Once the LCM is found, we adjust the fractions:

• Convert \( \frac{-4}{7} \) to \( \frac{-8}{14} \)
• Convert \( \frac{1}{2} \) to \( \frac{7}{14} \)

Now, both fractions share the common denominator of 14 and can be added: \[ \frac{-8}{14} + \frac{7}{14} = \frac{-1}{14} \]This step simplifies the equation, allowing us to solve for the variable \( x \). Understanding how to find and use a common denominator is essential for working with fractions effectively.