Problem 64 - Algebra for College Students

Question

Solve each equation, and check the solution. \(\frac{5-x}{6}+\frac{5}{6}=\frac{x}{54}\)

Step-by-Step Solution

Answer

x = 9

Step 1: Combine the fractions on the left-hand side

Combine the fractions on the left-hand side of the equation: \( \frac{5-x}{6} + \frac{5}{6} = \frac{5-x + 5}{6} \)

Step 2: Simplify the fraction

Simplify the numerator: \( \frac{10-x}{6} = \frac{x}{54} \)

Step 3: Cross multiply to clear the fractions

Multiply both sides by the denominators to clear the fractions: \( 54(10-x) = 6x \)

Step 4: Distribute and solve for x

Distribute 54 on the left-hand side: \( 540 - 54x = 6x \) Add 54x to both sides: \( 540 = 60x \) Divide by 60: \( x = 9 \)

Step 5: Check the solution

Substitute \( x = 9 \) back into the original equation to verify: \( \frac{5-9}{6} + \frac{5}{6} = \frac{9}{54} \) This simplifies to: \( \frac{-4}{6} + \frac{5}{6} = \frac{9}{54} \) Simplify the left-hand side: \( \frac{1}{6} = \frac{1}{6} \) The solution checks out.

Key Concepts

Cross Multiplication

Cross multiplication is a method used to simplify equations that contain fractions. It helps by eliminating the fractions, making the equation easier to solve. In our exercise, we have the equation \( \frac{10-x}{6} = \frac{x}{54} \). To clear the fractions, we cross multiply:

• Multiply the numerator of the first fraction by the denominator of the second fraction: \(54(10-x)\)
• Multiply the denominator of the first fraction by the numerator of the second fraction: \(6x\)

This gives us: \( 54(10-x) = 6x \). Now we have a simple equation without fractions, making it easier to solve for \(x\).

Combining Fractions

Combining fractions involves adding or subtracting fractions with a common denominator. In our exercise, we need to combine the fractions on the left-hand side of the equation: \( \frac{5-x}{6} + \frac{5}{6} \). Since both fractions have the same denominator (6), we can simply add the numerators together:

• First, write down the combined fraction: \( \frac{(5-x) + 5}{6} \)
• Simplify the numerator by combining like terms: \( \frac{10-x}{6} \)

Now, the fractions are combined into a single fraction: \( \frac{10-x}{6} \). This makes it easier to work with and solve later steps of the equation.

Checking Solutions

After solving an equation, it's always a good practice to check your solutions. This ensures that the obtained value satisfies the original equation. In our exercise, we found that \(x = 9\). To check if this is correct, substitute \(x = 9\) back into the original equation and verify:

• Original equation: \( \frac{5-9}{6} + \frac{5}{6} = \frac{9}{54} \)
• Simplify each fraction separately: \( \frac{-4}{6} + \frac{5}{6} \)
• Combine the fractions: \( \frac{1}{6} \)
• Compare: \( \frac{1}{6} = \frac{9}{54} \) which simplifies to \( \frac{1}{6} = \frac{1}{6} \)

The equation holds true, confirming that \( x = 9 \) is indeed the correct solution.