Math Problem Statement

Determine the constant of proportionality and write the direct variation equation for the ordered pairs (10, -2) and (90, -18).

Solution

To solve the problem, we need to determine the constant of proportionality (kk) for the given ordered pairs (10,2)(10, -2) and (90,18)(90, -18) and then express the direct variation equation.

Step 1: Formula for direct variation

The relationship for direct variation is given by: y=kxy = kx where kk is the constant of proportionality.

To find kk, we can use the formula: k=yxk = \frac{y}{x}

Step 2: Compute kk for each ordered pair

  1. For (10,2)(10, -2): k=210=0.2k = \frac{-2}{10} = -0.2

  2. For (90,18)(90, -18): k=1890=0.2k = \frac{-18}{90} = -0.2

Since kk is consistent for both points, the constant of proportionality is: k=0.2k = -0.2

Step 3: Write the direct variation equation

Using the formula y=kxy = kx and substituting k=0.2k = -0.2: y=0.2xy = -0.2x

Final Answer:

  • The constant of proportionality (kk) is 0.2-0.2.
  • The direct variation equation is: y=0.2xy = -0.2x

Would you like further details or examples?
Here are 5 related questions to deepen your understanding:

  1. How can we verify that the equation y=0.2xy = -0.2x holds for other values of xx?
  2. What are some real-life examples of direct variation?
  3. Can you identify the graph shape of a direct variation equation like y=0.2xy = -0.2x?
  4. What happens if k=0k = 0 in a direct variation?
  5. How do you solve for xx given yy in y=0.2xy = -0.2x?

Tip: Always double-check kk by substituting back into the equation with both ordered pairs!

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Math Problem Analysis

Mathematical Concepts

Direct Variation
Proportionality
Algebra

Formulas

y = kx
k = y/x

Theorems

Direct Variation Theorem

Suitable Grade Level

Grades 7-9