- Analyze the provided system: \(y = -6x + 2\) and \(-12x - 2y = -4\).
- Simplify the second equation by dividing by \(-2\) to get \(y = -6x + 2\).
- Compare the two equations: they are identical, which means they represent the same line.
- Conclude that the system has an infinite number of solutions.
- Final answer: \(\boxed{\text{infinite number of solutions}}\).
Explanation
- Analyzing the System of Linear Equations
We are given a system of two linear equations:
(1) \(y = -6x + 2\)
(2) \(-12x - 2y = -4\)
To determine the number of solutions, we can compare the two equations. A system of linear equations has infinitely many solutions if the two equations represent the same line, no solutions if they represent parallel and distinct lines, and exactly one solution if they represent intersecting lines.
- Transforming the Second Equation
Let's examine the second equation: \(-12x - 2y = -4\). To compare it with the first equation, we can rewrite it in the slope-intercept form, \(y = mx + b\).
First, divide every term in the equation \(-12x - 2y = -4\) by \(-2\):
\[\frac{-12x}{-2} - \frac{2y}{-2} = \frac{-4}{-2}\]
\[6x + y = 2\]
Next, subtract \(6x\) from both sides to isolate \(y\):
\[y = -6x + 2\]
- Comparing the Equations and Determining the Number of Solutions
Now we compare the transformed second equation, \(y = -6x + 2\), with the first equation, \(y = -6x + 2\).
Both equations are identical, meaning they represent the exact same line in the coordinate plane. Since every point on the line defined by the first equation satisfies the second equation, there are infinitely many points of intersection. Therefore, the system has an infinite number of solutions.
Examples
In real-world scenarios, such as in logistics or supply chain management, determining whether a system of equations has a unique solution or infinite solutions can signal whether a problem is fully constrained or has redundant information. For example, if you are modeling the cost of two different transportation routes and find that their cost equations are identical, it means any combination of variables that satisfies one route will automatically satisfy the other, indicating that you have flexibility in choosing your strategy within the defined constraints. This concept helps in identifying dependencies between different processes or variables.