3x - 4y = 12 \quad \text(1)\\ - Richter Guitar
Understanding the Line: 3x – 4y = 12 (1)
Solving the Equation, Graphing the Line, and Real-World Applications
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
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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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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.
