The coffee shop Java Jive has prices that increase linearly based on the size of the coffee. A 8 -ounce coffee costs $$ \$ 2.55 $$, and a 24 -ounce coffee costs $$ \$ 4.15 $$. Let $$ y $$ represent the cost, in dollars, of an $$ x $$ - ounce coffee. Write the equation representing this situation in point-slope form. Then write the equation in slope-intercept form. (Please use decimals instead of fractions in your answers.) point-slope form: slope-intercept form: Question Help:

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To model the pricing of coffee at Java Jive, we'll start by determining the relationship between the size of the coffee ($$ x $$ ounces) and its cost ($$ y $$ dollars). Since the prices increase linearly, the relationship can be expressed using a linear equation.

### Step 1: Calculate the Slope ($$ m $$)

The slope ($$ m $$) of the line represents the rate at which the cost increases per ounce of coffee. It's calculated using two given points:

$$
(8, 2.55) \quad \text{and} \quad (24, 4.15)
$$

$$
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4.15 - 2.55}{24 - 8} = \frac{1.60}{16} = 0.10
$$

So, the slope $$ m = 0.10 $$.

### Step 2: Write the Equation in Point-Slope Form

The point-slope form of a linear equation is:

$$
y - y_1 = m(x - x_1)
$$

Using the point $$ (8, 2.55) $$:

$$
y - 2.55 = 0.10(x - 8)
$$

### Step 3: Convert to Slope-Intercept Form

The slope-intercept form is:

$$
y = mx + b
$$

We already have $$ m = 0.10 $$. To find $$ b $$, substitute one of the points into the equation. Using $$ (8, 2.55) $$:

$$
2.55 = 0.10(8) + b \
2.55 = 0.80 + b \
b = 2.55 - 0.80 \
b = 1.75
$$

So, the slope-intercept form is:

$$
y = 0.10x + 1.75
$$

### **Final Answers:**

- **Point-Slope Form:**
  $$
  y - 2.55 = 0.10(x - 8)
  $$

- **Slope-Intercept Form:**
  $$
  y = 0.10\,x + 1.75
  $$
Point-slope form: $$ y - 2.55 = 0.10(x - 8) $$ Slope-intercept form: $$ y = 0.10x + 1.75 $$

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