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explicame paso a paso como resolver estos ejercicios

To help you solve these exercises step by step, I'll first outline what each problem requires and then go through the solution process.

### 1. **Given two points $ A(-1,5) $ and $ B(2,-8) $, find:**
   - **a. The slope of the line.**
   - **b. The equation of the line.**
   - **c. The graph of the line.**

### 2. **Find the monthly demand when the product price is $18.00, given:**
   - **a. The slope.**
   - **b. The linear equation.**

---

### **Step-by-Step Solutions**

#### **1. Given two points $ A(-1,5) $ and $ B(2,-8) $:**

**a. Find the slope of the line (m).**

The formula to calculate the slope between two points $ (x_1, y_1) $ and $ (x_2, y_2) $ is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

For the points $ A(-1,5) $ and $ B(2,-8) $:
- $ x_1 = -1 $, $ y_1 = 5 $
- $ x_2 = 2 $, $ y_2 = -8 $

Substitute these values into the formula:

\[
m = \frac{-8 - 5}{2 - (-1)} = \frac{-13}{3}
\]

So, the slope $ m = -\frac{13}{3} $.

**b. Find the equation of the line.**

The equation of a line in slope-intercept form is:

\[
y = mx + b
\]

Where:
- $ m $ is the slope.
- $ b $ is the y-intercept.

We already found the slope $ m = -\frac{13}{3} $.

To find $ b $, we can use one of the points (let's use point $ A(-1,5) $):

\[
y = mx + b \Rightarrow 5 = -\frac{13}{3}(-1) + b
\]

Solve for $ b $:

\[
5 = \frac{13}{3} + b \Rightarrow b = 5 - \frac{13}{3} = \frac{15}{3} - \frac{13}{3} = \frac{2}{3}
\]

So, the equation of the line is:

\[
y = -\frac{13}{3}x + \frac{2}{3}
\]

**c. Graph the line.**

To graph the line, you need to plot two points. You can use the points $ A(-1,5) $ and $ B(2,-8) $ or find the y-intercept $ \left(0, \frac{2}{3}\right) $. Connect these points to form the line.

#### **2. Linear Demand Problem**

**a. Find the slope.**

It seems this part requires you to calculate a slope based on given demand and price information, but the specific values are not clear from the text alone. 

**b. Find the linear equation.**

For a linear demand equation, the general form is:

\[
q = mp + b
\]

Where:
- $ q $ is the quantity demanded.
- $ p $ is the price.
- $ m $ is the slope (which represents how much quantity changes with price).
- $ b $ is the y-intercept (which represents the demand when the price is zero).

Given the information provided:
- At $ p = 0 $ (zero price), $ q = 26000 $.
- At $ p = 10 $, $ q = 10000 $.

We can use this information to calculate the slope:

\[
m = \frac{10000 - 26000}{10 - 0} = \frac{-16000}{10} = -1600
\]

The linear equation would therefore be:

\[
q = -1600p + 26000
\]

To find the demand when the price is $18.00:

\[
q = -1600(18) + 26000 = -28800 + 26000 = -2800
\]

It seems there may be an inconsistency in the values provided, but typically this would be how you calculate it.

---

Would you like more details on any of these steps or do you have further questions? Here are eight related questions to expand your understanding:

1. How do you interpret the slope in a real-world context like price and demand?
2. What does a negative slope indicate about the relationship between two variables?
3. How can you find the y-intercept using only two points?
4. What does the y-intercept represent in a demand equation?
5. How would the demand change if the slope were positive?
6. How do you graph a line with a fractional slope?
7. How can you verify your linear equation using additional points?
8. What other forms can the equation of a line take, and how do you convert between them?

**Tip:** When calculating the slope, always double-check your subtraction to avoid sign errors.

Explore detailed step-by-step solutions for linear equations and demand functions. Learn how to find slopes, graph lines, and calculate demand at specific prices. Suitable for high school students studying algebra and economics.

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