Verified Answer :

  • Identify the primary linear equation: \(501 = 578 - 3x\).
  • Isolate the variable term by rearranging the equation: \(501 - 578 = -3x\), which simplifies to \(-77 = -3x\).
  • Divide by \(-3\) to isolate \(x\): \(x = \frac{-77}{-3}\).
  • Obtain the final value of \(x\): \(x = \boxed{\frac{77}{3}}\).

Explanation

  1. Analysis of the Problem

We are presented with a set of equations and values: \(501 = 578 - 3x\), \(-\frac{71}{3x} = \frac{-3x}{-3x}\), and \(x = 25.67\). Upon analysis, these expressions are inconsistent with one another. The primary equation that allows us to solve for \(x\) in a standard algebraic manner is \(501 = 578 - 3x\). We will focus on solving this linear equation for \(x\).

  1. Isolating the Term with x

To solve for \(x\) in the equation \(501 = 578 - 3x\), we first isolate the term containing \(x\). We can do this by subtracting \(578\) from both sides of the equation:
\[501 - 578 = -3x\]
Simplifying the left side, we get:
\[-77 = -3x\]

  1. Solving for x

Now that we have \(-77 = -3x\), we solve for \(x\) by dividing both sides of the equation by \(-3\):
\[x = \frac{-77}{-3}\]
Since the quotient of two negative numbers is positive, we have:
\[x = \frac{77}{3}\]
As a decimal, this is approximately \(25.666...\), which rounds to \(25.67\). We will express the final answer as the fraction \(\frac{77}{3}\).

Examples

Linear equations like this are foundational in real-world scenarios such as calculating remaining balance on a loan. For example, if you owe \(\$578\) and pay \(\$501\) over a series of equal payments of \(\$3\) each, you can model the remaining debt with this type of linear equation. By solving for \(x\), you determine how many payments have been made or remain, allowing you to manage personal finances effectively.

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