Verified Answer :

  • Expand both sides of the equation: \(2x + 3x - 12 = 6 - 2x\).
  • Simplify the left side: \(5x - 12 = 6 - 2x\).
  • Move all terms involving \(x\) to one side by adding \(2x\): \(7x - 12 = 6\).
  • Isolate the term with \(x\) by adding 12 to both sides: \(7x = 18\).
  • Solve for \(x\): \(\boxed{\frac{18}{7}}\).

Explanation

  1. Expanding the Equation

To solve the linear equation \(2x + 3(x - 4) = 2(3 - x)\), we begin by expanding both sides of the equation. This involves applying the distributive property to eliminate the parentheses. On the left-hand side, we multiply 3 by both \(x\) and -4. On the right-hand side, we multiply 2 by both 3 and -x. This gives us:
\[2x + 3x - 12 = 6 - 2x\]

  1. Combining Like Terms

Next, we simplify the left-hand side by combining the like terms \(2x\) and \(3x\). Adding these together yields \(5x\), resulting in the simplified equation:
\[5x - 12 = 6 - 2x\]

  1. Isolating the Variable \(x\)

Now, we aim to isolate the variable \(x\) on one side of the equation. We add \(2x\) to both sides to move all terms containing \(x\) to the left:
\[5x + 2x - 12 = 6\]
\[7x - 12 = 6\]

  1. Solving for \(x\) constants

To further isolate \(x\), we add 12 to both sides of the equation:
\[7x = 6 + 12\]
\[7x = 18\]

  1. Final Calculation and Result

Finally, we divide both sides by 7 to solve for \(x\):
\[x = \frac{18}{7}\]
Thus, the solution to the equation is \(x = \frac{18}{7}\).

Examples

Linear equations are fundamental in many real-world scenarios, such as budgeting. For instance, if you have a base amount of money, a recurring expense, and a variable cost dependent on usage, you can model your total budget using a linear equation. Solving this equation helps determine how much you can afford to spend or how much you need to save to meet your financial goals. By simplifying and solving for the unknown variable, you can make informed decisions in everyday tasks like managing your allowance or planning a shopping trip.

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