Determine the demand equation assuming it is linear.
Understand the Problem
The question is asking to find the demand equation based on given quantities and prices, which involves using linear equations and concepts from algebra.
Answer
The demand equation is \( p = \frac{-7}{100}q + 65 \).
Answer for screen readers
The demand equation is given by:
$$ p = \frac{-7}{100}q + 65 $$
Steps to Solve
-
Identify the Points We have two points given:
((q_1, p_1) = (100, 58))
((q_2, p_2) = (200, 51)) -
Calculate the Slope (m) Use the slope formula:
$$ m = \frac{p_2 - p_1}{q_2 - q_1} $$
Substituting in the values:
$$ m = \frac{51 - 58}{200 - 100} = \frac{-7}{100} $$ -
Use the Point-Slope Form The point-slope formula is given by:
$$ p - p_1 = m(q - q_1) $$
Substituting one of the points (we'll use ((100, 58))) and the slope:
$$ p - 58 = \frac{-7}{100}(q - 100) $$ -
Simplify the Equation Distributing the slope:
$$ p - 58 = \frac{-7}{100}q + 7 $$
Now, solve for $p$:
$$ p = \frac{-7}{100}q + 65 $$
The demand equation is given by:
$$ p = \frac{-7}{100}q + 65 $$
More Information
This equation represents a linear relationship between the price (p) and quantity (q) of the product, indicating how the price decreases as more units are demanded.
Tips
- Forgetting to simplify the equation: after substituting values, always simplify to get the final demand equation.
- Incorrectly calculating the slope: Ensure correct subtraction and division in slope calculations.
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