3x - 4y = 12 \quad \text(1)\\ - Silent Sales Machine

Understanding the Line: 3x – 4y = 12 (1)
Solving the Equation, Graphing the Line, and Real-World Applications

The equation 3x – 4y = 12 (1) is a classic linear equation fundamental to algebra and geometry. Whether you're a student learning transformational geometry, a programmer working with coordinate systems, or someone trying to interpret real-world data, understanding how to manipulate and interpret this equation offers valuable insights. This article explores how to solve, graph, and apply the 3x – 4y = 12 (1) line in practical contexts.

Understanding the Context

What Is the Equation 3x – 4y = 12?

The equation 3x – 4y = 12 represents a straight line in two-dimensional space. It is expressed in standard form, where:

Ax + By = C

$3x - 4y = 12 \quad \text{(1)}\\$

Key Insights

In this case:

A = 3 (coefficient of x)
B = –4 (coefficient of y)
C = 12 (constant term)

Step 1: Solving for y in Terms of x (Slope-Intercept Form)

To better visualize and work with the line, we convert the equation into slope-intercept form:
y = mx + b

Starting with
3x – 4y = 12,
subtract 3x from both sides:
–4y = –3x + 12

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Final Thoughts

Now divide both sides by –4:
y = (3/4)x – 3

This reveals:

Slope (m) = 3/4 — meaning for every 4 units you move right, y increases by 3 units.
Y-intercept (b) = –3 — the line crosses the y-axis at the point (0, –3).

These values are critical for graphing and interpreting real-world trends.

Step 2: Finding the Intercepts

X-intercept: Set y = 0
3x – 4(0) = 12 → 3x = 12 → x = 4 → Point: (4, 0)

Y-intercept: Set x = 0
3(0) – 4y = 12 → –4y = 12 → y = –3 → Point: (0, –3)

Intercepts anchor the line on a graph, making it easier to plot and understand spatial relationships.