To determine a linear function that predicts the number of calculators sold per week based on the price $x$ (in dollars), we can set up a linear equation in the form:

$y = mx + b$

where:

• $y$ is the number of calculators sold per week,
• $x$ is the price per calculator,
• $m$ is the slope of the line, and
• $b$ is the y-intercept.

Step 1: Set up the points

From the information given:

1. At $x = 95$, $y = 10,000$.
2. At $x = 90$, $y = 12,000$.

Thus, we have two points: $(95, 10000)$ and $(90, 12000)$.

Step 2: Find the slope $m$

The slope $m$ is calculated by the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substitute the values from the points:

$m = \frac{12000 - 10000}{90 - 95} = \frac{2000}{-5} = -400$

So, the slope $m$ is $-400$.

Step 3: Use the point-slope form to find the equation

Now we can use one of the points, say $(95, 10000)$, and the slope $-400$ to find the y-intercept $b$ using the point-slope form $y = mx + b$:

$10000 = -400 \cdot 95 + b$

Calculate $-400 \cdot 95$:

$10000 = -38000 + b$

Solving for $b$:

$b = 10000 + 38000 = 48000$

Step 4: Write the final equation

The linear function that predicts the number of calculators sold per week at a price of $x$ dollars is:

$y = -400x + 48000$

Interpretation

• For each $1 increase in price, the number of calculators sold decreases by 400.
• When the price is $0 (theoretically), the model predicts 48,000 calculators sold, though this would not be practically realistic.

Would you like further explanation or examples?

Here are 5 related questions that may help extend your understanding:

1. How many calculators will be sold if the price is set at $85?
2. What price would result in selling exactly 15,000 calculators per week?
3. Can we determine the maximum price before sales drop to zero?
4. How would changes in manufacturing costs impact this linear demand function?
5. What if the sales response is not linear; how would the model change?

Tip: Linear models are often used in economics but remember that real-world demand often includes non-linear factors like seasonal trends or competitor influence.