Verified Answer :

  • Identify the least common denominator as 12 to eliminate fractions in the equation.
  • Multiply each term by 12, resulting in the linear form \(2(3y - 1) + (4y + 3) = 28\).
  • Simplify the linear equation to \(10y + 1 = 28\).
  • Solve for \(y\) to obtain the final value of \(\boxed{2.7}\).

Explanation

  1. Analysis of the Linear Equation

We are tasked with solving the linear equation:
\[\frac{3y - 1}{6} + \frac{4y + 3}{12} = \frac{7}{3}\]
To begin, we analyze the equation to determine the best approach. The equation contains fractions with denominators of 6, 12, and 3. The most efficient way to solve this is to eliminate the fractions by multiplying the entire equation by the least common denominator (LCD) of the fractions. The LCD of 6, 12, and 3 is 12.

  1. Clearing the Fractions

To clear the fractions, we multiply each term on both sides of the equation by 12:
\[12 \cdot \left(\frac{3y - 1}{6}\right) + 12 \cdot \left(\frac{4y + 3}{12}\right) = 12 \cdot \left(\frac{7}{3}\right)\]
Simplifying each term:

  • \(12 \div 6 = 2\), so the first term becomes \(2(3y - 1)\).
  • \(12 \div 12 = 1\), so the second term becomes \(1(4y + 3)\).
  • \(12 \div 3 = 4\), so the right-hand side becomes \(4 \cdot 7 = 28\).

The equation is now:
\[2(3y - 1) + (4y + 3) = 28\]

  1. Simplifying the Equation

Now we distribute the constant 2 across the terms in the first parenthesis:
\[6y - 2 + 4y + 3 = 28\]
Next, combine the like terms on the left side (grouping the \(y\) terms and the constant terms):
\[(6y + 4y) + (-2 + 3) = 28\]
\[10y + 1 = 28\]

  1. Final Solving for y

To isolate \(y\), subtract 1 from both sides of the equation:
\[10y = 28 - 1\]
\[10y = 27\]
Finally, divide by 10 to solve for \(y\):
\[y = \frac{27}{10} = 2.7\]
The final solution for \(y\) is \(\boxed{2.7}\).

Examples

This type of equation is commonly encountered in everyday budget balancing. For instance, if you have multiple recurring payments with different billing frequencies (such as bi-weekly and monthly), you might need to find an equivalent monthly rate to manage your finances. Solving equations like this helps you calculate the unknown variables, such as a missing monthly payment amount, ensuring your budget remains balanced and accurate.

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