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Let's break down the problem provided in the image:

We are asked to determine the equation of a linear function \( f(x) \) that:
1. Passes through the point \( (3, -4) \).
2. Satisfies the condition that when \( x \) increases by 2 units, the value of \( y \) increases by 3 units.

### Step 1: Use the slope condition
The second condition tells us that the slope \( m \) is the ratio of the change in \( y \) to the change in \( x \). Specifically:
\[
m = \frac{\Delta y}{\Delta x} = \frac{3}{2}
\]
So, the slope of the linear function is \( m = \frac{3}{2} \).

### Step 2: Write the general form of the linear function
The general form of a linear function is:
\[
f(x) = mx + b
\]
Substituting \( m = \frac{3}{2} \), we have:
\[
f(x) = \frac{3}{2}x + b
\]

### Step 3: Use the given point \( (3, -4) \)
Now, substitute \( x = 3 \) and \( f(x) = -4 \) into the equation to solve for \( b \):
\[
-4 = \frac{3}{2}(3) + b
\]
\[
-4 = \frac{9}{2} + b
\]
Multiply both sides by 2 to eliminate the fraction:
\[
-8 = 9 + 2b
\]
\[
2b = -8 - 9 = -17
\]
\[
b = -\frac{17}{2}
\]

### Step 4: Write the final equation
The equation of the linear function is:
\[
f(x) = \frac{3}{2}x - \frac{17}{2}
\]

### Step 5: Compare with the options
The correct answer is option **C**, which is:
\[
f(x) = \frac{3}{2}x - \frac{17}{2}
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you find the slope of a line given two points?
2. What is the significance of the slope in a linear equation?
3. How do you convert a linear equation to slope-intercept form?
4. What happens to the graph of a linear equation when you change the slope?
5. How do you verify if a point lies on a given line?

**Tip:** The slope-intercept form \( y = mx + b \) is useful when you know the slope and one point on the line.

Determine the equation of the linear function whose graph passes through the point (3, -4) and satisfies that when x increases by 2 units, y increases by 3 units.

Learn how to determine the equation of a linear function that passes through a specific point and meets a given slope condition. This problem covers key algebraic concepts, including slope and slope-intercept form, and is suitable for students in Grades 9-11.

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