Verified Answer :

  • Start with the original equation \(5x + 20y = 20\).
  • Subtract \(20y\) from both sides: \(5x = 20 - 20y\).
  • Divide every term by 5 to isolate \(x\): \(x = 4 - 4y\).
  • Match the result with the given options, identifying the final answer as \(\boxed{x = -4y + 4}\).

Explanation

  1. Problem Analysis and Objective

We are given the linear equation \(5x + 20y = 20\) and we need to isolate the variable \(x\). This process involves basic algebraic manipulation to express \(x\) in terms of \(y\). We want to arrive at an equation in the form \(x = my + b\), where \(m\) and \(b\) are constants.

  1. Step-by-Step Manipulation

To isolate \(x\), we first subtract the term containing \(y\) from both sides of the equation:
\[5x + 20y - 20y = 20 - 20y\]
\[5x = 20 - 20y\]
Now, to solve for \(x\), we divide both sides by the coefficient of \(x\), which is 5:
\[x = \frac{20 - 20y}{5}\]
Applying the division to each term on the right side:
\[x = \frac{20}{5} - \frac{20y}{5}\]
\[x = 4 - 4y\]
Rearranging the terms gives \(x = -4y + 4\).

  1. Comparing and Selecting the Correct Answer

Comparing our result, \(x = -4y + 4\), with the given options:

  • \(x = -2y + 4\)
  • \(x = \frac{3y + 8}{2}\)
  • \(x = \frac{-5y + 8}{2}\)
  • \(x = -4y + 4\)
  • \(x = 20\)

The correct option is \(x = -4y + 4\).

Examples

Linear equations are fundamental in budget management. For example, if you have a total of \(\$20\) to spend on two items, where the first costs \(\$5\) per unit (\(x\)) and the second costs \(\$20\) per unit (\(y\)), your spending is represented by \(5x + 20y = 20\). By solving for \(x\), you can quickly determine how many units of the first item you can afford for any given amount of the second item purchased. This helps in making informed financial decisions.

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