To sketch the graph of the equation \(3x - 4y = 12\), we’ll follow these steps:

### 1. Rewrite the Equation in Slope-Intercept Form
Rewrite \(3x - 4y = 12\) as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept:

\[
3x - 4y = 12
\]
Subtract \(3x\) from both sides:
\[
-4y = -3x + 12
\]
Now, divide every term by \(-4\):
\[
y = \frac{3}{4}x - 3
\]

So, the equation in slope-intercept form is:
\[
y = \frac{3}{4}x - 3
\]

This means:
- The slope (\(m\)) is \(\frac{3}{4}\), indicating the line rises 3 units for every 4 units it moves to the right.
- The y-intercept (\(b\)) is \(-3\), so the line crosses the y-axis at \((0, -3)\).

### 2. Identify Key Points
To draw the line accurately, find at least two points on the graph:

- **Y-intercept**: When \(x = 0\), \(y = -3\).
- **X-intercept**: Set \(y = 0\) and solve for \(x\):
  \[
  3x - 4(0) = 12
  \]
  \[
  x = 4
  \]
  So, the x-intercept is \((4, 0)\).

### 3. Plot the Points and Draw the Line
1. Plot the y-intercept \((0, -3)\) on the coordinate plane.
2. Plot the x-intercept \((4, 0)\) as well.
3. Draw a straight line through these two points to represent the equation.

This is the graph of the equation \(3x - 4y = 12\).

Would you like me to generate an image of this graph?

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Here are 5 related questions for further understanding:
1. How can the slope of a line indicate the steepness and direction of the line?
2. What is the purpose of converting a linear equation to slope-intercept form?
3. How can we verify a point lies on the graph of an equation?
4. What does changing the y-intercept in an equation do to the graph?
5. How can we find the equation of a line if we only know two points on it?

**Tip:** When graphing, always label your intercepts and key points for clarity.

Learn how to sketch the graph of the linear equation 3x - 4y = 12 with a detailed, step-by-step solution. This guide explains how to convert the equation to slope-intercept form, identify the x- and y-intercepts, and plot the line. Ideal for students in Grades 8-10.

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