Problem 16 - Elementary and Intermediate Algebra

Question

Solve each equation and check the result. If an equation has no solution, so indicate. $$ \frac{7}{4}=\frac{x}{8}+\frac{5}{2} $$

Step-by-Step Solution

Answer

The solution is \( x = -6 \).

Step 1: Isolate the Variable Term

Start with the equation \( \frac{7}{4} = \frac{x}{8} + \frac{5}{2} \). The objective is to isolate the \( \frac{x}{8} \) term. Subtract \( \frac{5}{2} \) from both sides to get:\[ \frac{7}{4} - \frac{5}{2} = \frac{x}{8} \]

Step 2: Simplify the Left Side

To subtract fractions, they need a common denominator. The denominators are 4 and 2, with a common denominator of 4. Rewrite \( \frac{5}{2} \) as \( \frac{10}{4} \):\[ \frac{7}{4} - \frac{10}{4} = \frac{x}{8} \] Simplify the left side:\[ \frac{-3}{4} = \frac{x}{8} \]

Step 3: Solve for x

To solve for \( x \), multiply both sides by 8 to get rid of the fraction on the right side:\[ 8 \times \frac{-3}{4} = x \]Simplify the left side:\[ -6 = x \]

Step 4: Verify the Solution

Substitute \( x = -6 \) back into the original equation to verify:\[ \frac{7}{4} = \frac{-6}{8} + \frac{5}{2} \]First, simplify \( \frac{-6}{8} \) to \( \frac{-3}{4} \). Then convert \( \frac{5}{2} \) to \( \frac{10}{4} \). Add the fractions on the right side:\[ \frac{-3}{4} + \frac{10}{4} = \frac{7}{4} \]Since both sides of the equation are equal, \( x = -6 \) is the correct solution.

Key Concepts

Isolating the Variable

When solving equations, the goal is to determine the value of the unknown variable, often represented by \( x \). In the equation \( \frac{7}{4} = \frac{x}{8} + \frac{5}{2} \), the focus is to isolate the term containing \( x \). This means you need to move everything else to the other side of the equation. By doing so, you make \( x \) the focal point of the equation.

Start by eliminating any numbers or terms that complicate the expression with \( x \) in it. In this example, subtract \( \frac{5}{2} \) from both sides, as shown:

• Initial equation: \( \frac{7}{4} = \frac{x}{8} + \frac{5}{2} \)
• Subtract \( \frac{5}{2} \) from both sides: \( \frac{7}{4} - \frac{5}{2} = \frac{x}{8} \)

Now, \( x \) is isolated on one side, allowing for easier manipulation and next steps towards finding its value.

Common Denominator

To work with fractions effectively, especially when adding or subtracting, it’s crucial to have a common denominator. It helps in combining fractions without distorting their values.
In our equation \( \frac{7}{4} - \frac{5}{2} = \frac{x}{8} \), we need to subtract \( \frac{5}{2} \) from \( \frac{7}{4} \). However, their denominators are different. Here's how you find a common denominator:

• Denominators are 4 and 2. The least common multiple is 4.
• Convert \( \frac{5}{2} \) to \( \frac{10}{4} \).

Now you can easily subtract:

• \( \frac{7}{4} - \frac{10}{4} = \frac{-3}{4} \)

This method ensures the operations involving fractions remain precise and accurate, avoiding any mishaps with the solution.

Checking Solutions

Once you find a solution to an equation, like \( x = -6 \), it’s essential to confirm its accuracy. You do this by substituting the solution back into the original equation to see if it upholds the equation's truth. Verification minimizes mistakes and assures the solution is correct.

Here’s how you check the solution \( x = -6 \):

• Original equation: \( \frac{7}{4} = \frac{x}{8} + \frac{5}{2} \)
• Replace \( x \) with \(-6\): \( \frac{7}{4} = \frac{-6}{8} + \frac{5}{2} \)
• Simplify \( \frac{-6}{8} \) to \( \frac{-3}{4} \)
• Convert \( \frac{5}{2} \) to \( \frac{10}{4} \)
• Combine: \( \frac{-3}{4} + \frac{10}{4} = \frac{7}{4} \)

Both sides equate to \( \frac{7}{4} \), confirming that \( x = -6 \) is indeed the correct solution. This final check is crucial to ensure the validity of the result obtained.