Verified Answer :

  • The given linear equation is \(3x + 7 = 4x - 5\).
  • Subtract \(3x\) from both sides to get \(7 = x - 5\).
  • Add \(5\) to both sides to isolate the variable, yielding \(x = 12\).
  • The solution to the equation is \(\boxed{12}\).

Explanation

  1. Analysis of the equation

We are given the linear equation \(3x + 7 = 4x - 5\). Our objective is to find the value of \(x\) that satisfies this equation. A linear equation of this form contains both variable terms and constant terms on both sides of the equation. Our strategy is to isolate the variable \(x\) by grouping all terms containing \(x\) on one side of the equation and all constant terms on the other side.

  1. Isolating the variable \(x\)

To isolate the variable, we first subtract \(3x\) from both sides of the equation to collect the \(x\)-terms on one side:
\[3x - 3x + 7 = 4x - 3x - 5\]
\[7 = x - 5\]
Next, we add \(5\) to both sides of the equation to isolate the \(x\)-term:
\[7 + 5 = x - 5 + 5\]
\[12 = x\]
Thus, we have found that \(x = 12\).

  1. Verification and conclusion

To verify our result, we substitute \(x = 12\) back into the original equation \(3x + 7 = 4x - 5\):
Left-hand side: \(3(12) + 7 = 36 + 7 = 43\).
Right-hand side: \(4(12) - 5 = 48 - 5 = 43\).
Since both sides result in \(43\), our solution is correct. The final value is \(\boxed{12}\).

Examples

Linear equations are foundational in many everyday situations. For instance, imagine comparing two pricing plans for a service. Plan A charges a flat fee of \(\$7\) plus \(\$3\) per hour, represented by \(3x + 7\). Plan B charges a flat fee of \(-\$5\) (perhaps a discount) plus \(\$4\) per hour, represented by \(4x - 5\). Setting them equal, \(3x + 7 = 4x - 5\), allows us to find the exact number of hours, \(x = 12\), where both plans cost the same amount, \(\$43\). This helps in making informed decisions about which plan to choose based on usage.

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