- Identify the condition for infinite solutions: the equations must represent the same line.
- Transform the given equation \(3y = 2x - 9\) into slope-intercept form: \(y = \frac{2}{3}x - 3\).
- Compare the given options to this form, finding that \(y = \frac{2}{3}x - 3\) matches.
- Conclude that the other equation is \(\boxed{y = \frac{2}{3}x - 3}\).
Explanation
- Analyzing the Conditions for Infinite Solutions
We are given a system of two linear equations and told that it has an infinite number of solutions. The first equation is \(3y = 2x - 9\). For a system of two linear equations to have an infinite number of solutions, the two equations must be equivalent, meaning they represent the same line. This occurs if one equation is a non-zero constant multiple of the other. Our goal is to identify which of the provided options is equivalent to the given equation \(3y = 2x - 9\).
- Standardizing the Given Equation
First, let us simplify the given equation \(3y = 2x - 9\) into slope-intercept form (\(y = mx + b\)). By dividing both sides of the equation by 3, we get:
\[y = \frac{2}{3}x - \frac{9}{3} = \frac{2}{3}x - 3.\]
The slope of this line is \(m = \frac{2}{3}\) and the y-intercept is \(b = -3\). Two equations represent the same line if and only if they have the same slope and y-intercept when written in slope-intercept form, or if one is a constant multiple of the other in their general forms.
- Evaluating the Options
Now, we evaluate each of the given options to see which one represents the same line as \(y = \frac{2}{3}x - 3\):
- Option 1: \(2y = x - 4.5 \implies y = \frac{1}{2}x - 2.25\). (Different slope and intercept)
- Option 2: \(y = \frac{2}{3}x - 3\). (Matches exactly)
- Option 3: \(6y = 6x - 27 \implies y = x - 4.5\). (Different slope and intercept)
- Option 4: \(y = \frac{3}{2}x - 4.5\). (Different slope and intercept)
Comparing these, only \(y = \frac{2}{3}x - 3\) is identical to our original equation. Therefore, it is the only equation that, when paired with the original, forms a system with an infinite number of solutions.
- Conclusion
Since the second equation must represent the same line as the first to have an infinite number of solutions, we have confirmed that the second equation must be \(y = \frac{2}{3}x - 3\). Thus, the correct choice is the second one.
Examples
In urban planning, if you are mapping out two different public transit lines that overlap exactly, they represent the same path on the map. Solving systems of equations helps determine if two proposed route functions describe the same set of coordinates. By converting equations to the slope-intercept form, planners can quickly verify if the routes will coincide, ensuring there is no confusion in the network layout.