Verified Answer :

  • Combine the coefficients of \(x\) on the left side: \(-3.8x - 5.9x = -9.7x\).
  • Set the simplified equation to the constant on the right: \(-9.7x = 223.1\).
  • Divide by \(-9.7\) to solve for \(x\): \(x = \frac{223.1}{-9.7}\).
  • The final result is \(x = \boxed{-23}\).

Explanation

  1. Analyzing the Equation

We are tasked with solving for the variable \(x\) in the linear equation \(-3.8x - 5.9x = 223.1\). The equation contains two terms on the left-hand side that share the same variable \(x\), making it a simple linear equation. Our primary objective is to isolate \(x\) to find its value.

  1. Combining Like Terms

The first step in solving this equation is to combine the coefficients of \(x\) on the left-hand side. We have:
\[-3.8x - 5.9x = (-3.8 - 5.9)x\]
Calculating the sum of the coefficients:
\[-3.8 - 5.9 = -9.7\]
So, the equation simplifies to:
\[-9.7x = 223.1\]

  1. Isolating the Variable and Final Calculation

Now that the equation is in the form \(Ax = B\), where \(A = -9.7\) and \(B = 223.1\), we can isolate \(x\) by dividing both sides of the equation by the coefficient \(-9.7\):
\[x = \frac{223.1}{-9.7}\]
Performing the division:
\[x = -23\]
Thus, the value of \(x\) is \(-23\).

Examples

Linear equations are foundational in balancing budgets or predicting outcomes. Imagine you are tracking the total change in your bank account balance over two transactions. If one transaction is a withdrawal of \(\$3.80\) per unit of something and another is a withdrawal of \(\$5.90\) per unit, and your total balance change is \(\$223.10\) (represented as a negative total withdrawal), this equation helps determine how many units were processed. Understanding how to group and solve these coefficients ensures accurate financial planning and resource management.

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