Problem 22 - Algebra for College Students

Question

Solve each equation. \(\frac{y}{3}+\frac{y-2}{8}=\frac{6 y-1}{12}\)

Step-by-Step Solution

Answer

The solution is \(y = -4\).

Step 1: Identify the Least Common Denominator (LCD)

To solve the equation, look for the least common denominator (LCD) of all the fractions. The denominators are 3, 8, and 12. The LCD for these denominators is 24.

Step 2: Clear the Fractions

Multiply every term in the equation by the LCD (24) to eliminate the fractions:\[24 \times \left( \frac{y}{3} \right) + 24 \times \left( \frac{y-2}{8} \right) = 24 \times \left( \frac{6y-1}{12} \right)\] Simplifying this gives:\[8y + 3(y - 2) = 2(6y - 1)\]

Step 3: Simplify the Equation

Distribute the terms:\[8y + 3y - 6 = 12y - 2\] Combine like terms:\[11y - 6 = 12y - 2\]

Step 4: Isolate the Variable 'y'

Subtract 12y from both sides to get:\[11y - 12y - 6 = -2\] Simplifies to:\[-y - 6 = -2\]

Step 5: Solve for 'y'

Add 6 to both sides:\[-y = 4\] Multiply both sides by -1 to get:\[y = -4\]

Key Concepts

Least Common Denominator

When dealing with fractional equations, it's crucial to find a common denominator to fairly combine or compare the fractions. The least common denominator (LCD) is simply the smallest number that the denominators of all fractions in the equation can evenly divide into. For instance, in the equation \(\frac{y}{3}+\frac{y-2}{8}=\frac{6y-1}{12}\), the denominators are 3, 8, and 12.

To find the LCD:

• List the multiples for each denominator: 3 (3, 6, 9, 12, 15, 18, 21, 24...), 8 (8, 16, 24, 32...), and 12 (12, 24, 36...).
• Identify the smallest common multiple: 24.

This LCD of 24 means you can rewrite each fraction with this common base, making it possible to eliminate the fractions from the equation later on by multiplying each term by this number.

Fraction Elimination

Eliminating fractions from your equation simplifies your work and leads to easier manipulation. By multiplying every term in an equation by the LCD, you remove fractions, thus turning it into a simpler linear equation.

Let's look at our example: After finding that the LCD is 24, multiply each term:

• \(24 \times \left( \frac{y}{3} \right) = 8y\)
• \(24 \times \left( \frac{y-2}{8} \right) = 3(y-2)\)
• \(24 \times \left( \frac{6y-1}{12} \right) = 2(6y-1)\)

By doing this, we effectively clear the equation of fractions, preparing it for further simplification and variable isolation.

Variable Isolation

To find the value of the variable, you must isolate it on one side of the equation. Start by simplifying the equation to gather all terms involving the variable on one side and constant terms on the other.

With our simplified equation \(8y + 3(y-2) = 2(6y-1)\), distribute and combine like terms:

• Distribute the constants, resulting in \(8y + 3y - 6\) and \(12y - 2\)
• Combine like terms to get \(11y - 6 = 12y - 2\)

Next, move the variable terms to one side by subtracting \(12y\) from both sides, resulting in \(-y - 6 = -2\). Finish up by getting the variable \(y\) alone.

Simplifying Equations

Simplifying equations means making them as manageable as possible by reducing to the smallest form. Here we already have the equation \(-y - 6 = -2\). Follow these steps to complete the simplification:

• To eliminate the \(-6\), add 6 to both sides resulting in \(-y = 4\).
• To solve for \(y\), multiply both sides by \(-1\) to turn \(-y\) into \(y\), yielding \(y = -4\).

By doing this, the equation is simplified and the variable isolated, providing a straightforward solution, \(y = -4\), to the original problem. Keep your steps organized for clarity and verification purposes.