Problem 44 - Algebra 1: Concepts and Skills

Question

MULTIPLE CHOICE Assuming \(y=9\) when \(x=10,\) find an equation that relates \(x\) and \(y\) such that \(x\) and \(y\) vary inversely. $$ F) x y=90 $$ $$ G)y=\frac{9}{10} x $$ $$ H)y=\frac{10}{9} x $$ $$ J)x y=\frac{9}{10} $$

Step-by-Step Solution

Answer

The equation that relates \(x\) and \(y\) such that they vary inversely is \(x \cdot y = 90\). Therefore, the correct choice is F) \(x \cdot y = 90\)

Step 1: Understand the inverse relationship

Begin by understanding the basis of inverse variation. In an inverse variation, the product \(x \cdot y\) is equal to a constant \(k\): \(x \cdot y = k\). Here, \(k\) is a constant that does not change.

Step 2: Substitute given values for \(x\) and \(y\) in the inverse relationship

Next, substitute the provided values for \(x\) and \(y\), that is, \(x=10\) and \(y=9\). When you substitute these values in the equation \(x \cdot y = k\), we get \(10 \cdot 9 = k\), which computes to \(90 = k\). Thus, the constant \(k\) equals \(90\) for the given values of \(x\) and \(y\).

Step 3: Find the equation that relates \(x\) and \(y\)

Now, using the derived value of constant \(k\), form the equation that involves \(x\) and \(y\). That equation is \(x \cdot y = 90\).

Step 4: Identify the correct choice

Lastly, compare the equation derived with the multiple choice options given in the question. Option F) \(x \cdot y = 90\) is the correct choice.

Key Concepts

Constant of Variation

The constant of variation is a special value in inverse variation problems. It is a constant that shows the consistent relationship between two variables.

In any inverse variation, when one variable increases, the other decreases, keeping the product of the two values the same.
For instance, in the exercise, the product of \(x\) and \(y\) is always equal to 90. This value is known as the constant of variation, often represented by \(k\).

Understanding this idea helps:

• Predict how changes in one variable affect the other.
• Determine relationships within algebraic formulas.

Recognizing that \(k\) will always stay the same makes these types of problems predictable and easier to solve.

Algebraic Equations

Algebraic equations are formulas that use numbers, variables, and operations to express mathematical relationships.

In an inverse variation, these equations are in the form \(x \cdot y = k\). Here, variables \(x\) and \(y\) fluctuate together, ensuring their product remains the constant \(k\).

The equation \(x \cdot y = 90\) is crucial to our problem. It relates \(x\) and \(y\) in such a way that as one increases, the other decreases proportionally to maintain the constant \(90\).

Ensuring you understand this relationship can help you:

• Form accurate representations through equations in your maths problems.
• Relate variables in real-life scenarios where inverse relationships exist like speed and time.

These equations often serve as the foundation of solving real-world scenarios that involve relationships under constant terms.

Problem Solving Steps

Solving inverse variation problems can be grasped through a consistent approach.

Start by identifying your constant of variation \(k\). This is done by knowing the product format \(x \cdot y = k\).

Follow these problem-solving steps:

• **Identify**: Recognize the form of the equation and identify the inverse relationship.
• **Substitute Values**: Put the given numbers into the equation, like substituting \(x=10\) and \(y=9\) to find \(k\).
• **Solve for k**: Calculate \(k\) to understand the constant. In our case, \(10 \cdot 9 = 90\).
• **Write the Equation**: Use the \(k\) discovered to write the solution equation, here it's \(x \cdot y = 90\).
• **Verify with Options**: Finally, compare your solution with available options to confirm accuracy, choosing the correct equation which is F) \(x \cdot y = 90\).

Adopting a structured approach simplifies problem-solving and fosters independence when solving mathematical problems.