- Identify the original line by converting \(3y = 2x - 9\) into slope-intercept form: \(y = \frac{2}{3}x - 3\).
- Recognize that an infinite number of solutions exists if the second equation is equivalent to the first (same slope and intercept).
- Compare each option against the slope-intercept form \(y = \frac{2}{3}x - 3\).
- Conclude that the second option \(y = \frac{2}{3}x - 3\) represents the same line.
- The final answer is \(\boxed{y = \frac{2}{3}x - 3}\).
Explanation
- Analyzing the original equation
The given equation is \(3y = 2x - 9\). To understand its behavior and identify equivalent equations, we first convert it to the slope-intercept form, \(y = mx + b\). Dividing both sides by 3, we obtain:
\[y = \frac{2}{3}x - 3.\]
This indicates that any equation representing the same line must also have a slope of \(\frac{2}{3}\) and a y-intercept of \(-3\). A system of two linear equations has an infinite number of solutions if and only if the two equations are equivalent, meaning they represent the same line. Mathematically, this occurs when one equation is a non-zero constant multiple of the other.
- Evaluating the options
We examine each of the provided options to see if they are equivalent to the original equation \(3y = 2x - 9\):
- Option 1: \(2y = x - 4.5\). Converting to slope-intercept form: \(y = \frac{1}{2}x - 2.25\). This does not match \(y = \frac{2}{3}x - 3\).
- Option 2: \(y = \frac{2}{3}x - 3\). Converting this to the original form by multiplying by 3: \(3y = 2x - 9\). This is identical to the original equation, representing the same line.
- Option 3: \(6y = 6x - 27\). Dividing by 6: \(y = x - 4.5\). This does not match.
- Option 4: \(y = \frac{3}{2}x - 4.5\). This has a different slope of \(\frac{3}{2}\) and does not match.
Therefore, the equation \(y = \frac{2}{3}x - 3\) is the only one equivalent to the original equation, ensuring the system has infinitely many solutions.
Examples
In fields like civil engineering or urban planning, equations are used to model the trajectories of two different transit lines. If an engineer finds that two different planned lines are actually represented by the same equation (e.g., in a coordinate system representing a map), it means the lines are collinear and overlap completely. Recognizing such systems helps in identifying redundant plans, ensuring that infrastructure designs are distinct and correctly mapped across the planned space.