Verified Answer :

  • Simplify the left-hand side by distributing the constants and combining like terms, resulting in \(y - 6\).
  • Simplify the right-hand side by expanding and observing that the variable terms cancel out, leaving the constant 22.
  • Solve the simplified linear equation \(y - 6 = 22\) by adding 6 to both sides.
  • Final answer: \(y = \boxed{28}\).

Explanation

  1. Analysis and Initial Expansion

To solve the linear equation \[6(9y - 1) - 10(5y) - 3y = 22 - 4(2y - 12) + 8(y - 6),\] we begin by expanding the terms within parentheses on both sides of the equation. Our goal is to simplify each side independently to isolate the variable \(y\).

  1. Simplification of Both Sides

First, focus on the left-hand side (LHS):
\[6(9y - 1) - 10(5y) - 3y = 54y - 6 - 50y - 3y.\]
Combining the like terms involving \(y\):
\[(54 - 50 - 3)y - 6 = y - 6.\]
Now, focus on the right-hand side (RHS):
\[22 - 4(2y - 12) + 8(y - 6) = 22 - 8y + 48 + 8y - 48.\]
Notice that the \(-8y\) and \(+8y\) terms cancel each other out, and the \(+48\) and \(-48\) constants cancel out as well:
\[22 + (-8y + 8y) + (48 - 48) = 22.\]
After simplification, the equation becomes:
\[y - 6 = 22.\]

  1. Solving for y

Now that the equation is simplified to \(y - 6 = 22\), we isolate the variable \(y\) by adding 6 to both sides of the equation:
\[y = 22 + 6 = 28.\]
The value of \(y\) is 28.

Examples

Linear equations are frequently used in budgeting and resource planning. For example, if you are planning a party and have a fixed amount of money that must cover different costs, such as a base venue fee plus a per-person charge for food and entertainment, you can model the total cost using a linear equation. If you know the total budget and the individual rates, solving the equation for the number of guests allows you to determine how many people can be invited while staying within your financial limits. This demonstrates how algebraic manipulation helps in making informed, real-world decisions.

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