- Identify the slope of the original equation \(2y - 4x = 12\) by rewriting it as \(y = 2x + 6\).
- Find the slope of each option:
- \(-y - 2x = 6 \implies y = -2x - 6\) (Slope = -2)
- \(-y + 2x = 12 \implies y = 2x - 12\) (Slope = 2)
- \(y = 2x + 6\) (Slope = 2)
- \(y = 2x + 12\) (Slope = 2)
- Recognize that a system of two lines has exactly one solution if their slopes are different.
- Conclude that \(-y - 2x = 6\) is the correct choice because its slope of -2 differs from the original slope of 2.
- The final answer is \(\boxed{-y - 2x = 6}\).
Explanation
- Analysis of the Problem
To determine which equation forms a system with exactly one solution when paired with the original equation \(2y - 4x = 12\), we first need to understand the relationship between linear equations and their solutions. A system of two linear equations has exactly one solution if and only if the two lines intersect at a single point. This occurs when the lines are not parallel, which means their slopes must be different. If the lines have the same slope, they are either parallel (zero solutions) or coincident (infinitely many solutions).
- Identifying the Slope of the Original Equation
First, let's rewrite the given equation, \(2y - 4x = 12\), in slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
Starting with \(2y - 4x = 12\), we add \(4x\) to both sides:
\[2y = 4x + 12\]
Then, dividing everything by 2:
\[y = 2x + 6\]
So, the slope \(m\) of the original equation is 2. For another equation to form a system with exactly one solution, it must have a slope different from 2.
- Comparing the Slopes of the Options
Now, let's analyze each of the given options by writing them in slope-intercept form and finding their slopes:
- \(-y - 2x = 6 \implies -y = 2x + 6 \implies y = -2x - 6\). The slope is \(-2\).
- \(-y + 2x = 12 \implies -y = -2x + 12 \implies y = 2x - 12\). The slope is 2.
- \(y = 2x + 6\). The slope is 2.
- \(y = 2x + 12\). The slope is 2.
We can see that only the first option, \(-y - 2x = 6\), has a slope of \(-2\), which is different from the slope of 2 in our original equation. The other three equations have a slope of 2, which means they are parallel to the original line (and since they have different intercepts, they are distinct, resulting in zero solutions or, in the case of the third option, they are the same line resulting in infinite solutions). Therefore, \(-y - 2x = 6\) is the only equation that will result in a single intersection point.
- Conclusion and Final Answer
We have determined that the original equation \(2y - 4x = 12\) has a slope of 2, and the option \(-y - 2x = 6\) has a slope of -2. Since their slopes are not equal, these two lines must intersect at exactly one point. Therefore, the equation that forms a system with exactly one solution is \(-y - 2x = 6\).
Examples
In real-life scenarios like logistics, managing delivery routes can be modeled using systems of equations. Imagine two vehicles starting at different locations and moving along paths defined by linear equations. If their paths are represented by lines with different slopes, they will inevitably cross at a specific point, which could represent a rendezvous point or a shared stop in their routes. Determining the intersection point helps logistics managers optimize scheduling and ensure both vehicles can meet at a designated time and location.