Verified Answer :

  • The expression is defined by the equation \(40 \div x = y\), where \(x\) is the divisor and \(y\) is the quotient.
  • The variables are related by the equation \(xy = 40\), which implies \(y = \frac{40}{x}\) for \(x eq 0\).
  • Infinite pairs of \((x, y)\) satisfy this relationship, such as \((1, 40), (2, 20), (4, 10), (5, 8)\), and \((8, 5)\).
  • The general solution is given by the set of all pairs \(\boxed{(x, \frac{40}{x}) \text{ for } x eq 0}\).

Explanation

  1. Understanding the Relationship Between Dividend, Divisor, and Quotient

The expression \(40 \div \_\_\_\_ = \_\_\_\_\) represents a division operation where the dividend is 40. Let the divisor be \(x\) and the quotient be \(y\). The equation can be written as:
\[40 \div x = y\]
where \(x\) is any non-zero real number. In a division problem, the dividend, divisor, and quotient are related by the identity:
\[\text{Dividend} = \text{Divisor} \times \text{Quotient}\]
Substituting our values, this means \(40 = x \times y\). Since we have one equation and two unknowns (\(x\) and \(y\)), there are infinitely many pairs \((x, y)\) that satisfy this relationship. To find a specific pair, we simply choose a non-zero value for \(x\) and calculate the corresponding value for \(y\).

  1. Generating Potential Solutions

To generate possible solutions for the equation \(40 \div x = y\), we can select various values for the divisor \(x\) and compute the corresponding quotient \(y\):

  • If we choose \(x = 1\), then \(y = 40 \div 1 = 40\). The pair is \((1, 40)\).
  • If we choose \(x = 2\), then \(y = 40 \div 2 = 20\). The pair is \((2, 20)\).
  • If we choose \(x = 4\), then \(y = 40 \div 4 = 10\). The pair is \((4, 10)\).
  • If we choose \(x = 5\), then \(y = 40 \div 5 = 8\). The pair is \((5, 8)\).
  • If we choose \(x = -4\), then \(y = 40 \div (-4) = -10\). The pair is \((-4, -10)\).

Any non-zero real number \(x\) will yield a valid quotient \(y = \frac{40}{x}\).

  1. Conclusion and Graphical Interpretation

The relationship \(y = \frac{40}{x}\) describes an inverse proportionality, which is represented visually by a hyperbola. Because there are infinite solutions, the set of all possible pairs \((x, y)\) forms this curve on a coordinate plane. The curve exists in the first quadrant for positive \(x\) values and in the third quadrant for negative \(x\) values. The equation has no solution at \(x=0\) because division by zero is undefined. Therefore, the set of all possible solutions is represented by the function \(y = \frac{40}{x}\) for all \(x \in \mathbb{R}, x eq 0\).

Examples

Suppose you are organizing a school fundraiser and you have 40 stickers to distribute equally among a group of students. The equation \(40 \div x = y\) helps you determine how many students (\(x\)) each receive a specific number of stickers (\(y\)). If you decide each student gets 5 stickers, you solve \(40 \div x = 5\), which gives \(x = 8\) students. If you decide to give them to 10 students, each student gets \(40 \div 10 = 4\) stickers. This simple algebraic relationship is a fundamental tool for planning and resource allocation in everyday life.

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