- Analyze the given equations: \(y = -6x + 2\) and \(-12x - 2y = -4\).
- Simplify the second equation by dividing by \(-2\), resulting in \(y = -6x + 2\).
- Recognize that both equations describe the identical line.
- Conclude that the system has an \(\boxed{\text{infinite number of solutions}}\).
Explanation
- Analyzing the System of Linear Equations
We are given the following system of two linear equations in two variables, \(x\) and \(y\):
\[ \begin{aligned} 1) \quad y &= -6x + 2 \ 2) \quad -12x - 2y &= -4 \end{aligned} \]
Our objective is to determine how many solutions this system has. A system of linear equations in two variables can either have one unique solution (if the lines intersect), no solution (if the lines are parallel and distinct), or an infinite number of solutions (if the lines are identical). To classify the relationship between these two equations, we can express them in the same form, such as the slope-intercept form \(y = mx + b\).
- Comparing the Equations
We already have the first equation in slope-intercept form: \(y = -6x + 2\). Now, let us manipulate the second equation to see if it can be written in the same form:
\[-12x - 2y = -4\]
To isolate \(y\), we first add \(12x\) to both sides:
\[-2y = 12x - 4\]
Next, divide every term by \(-2\):
\[y = \frac{12x}{-2} - \frac{4}{-2}\]
Simplifying this, we get:
\[y = -6x + 2\]
We observe that the second equation is algebraically equivalent to the first one. Both equations describe exactly the same line in the Cartesian coordinate plane.
- Conclusion on the Number of Solutions
Since both equations represent the same line, any point \((x, y)\) that satisfies \(y = -6x + 2\) will also satisfy the second equation. For example, if we pick \(x = 0\), then \(y = 2\). If we pick \(x = 1\), then \(y = -6(1) + 2 = -4\). Since there are infinitely many values for \(x\), there are infinitely many pairs of \((x, y)\) that satisfy both equations simultaneously. Therefore, the system has an infinite number of solutions.
Examples
In real life, systems of equations help us understand scenarios where multiple conditions must be met simultaneously. For instance, consider two different subscription plans for a music streaming service: one might charge a fixed monthly fee plus a per-song cost, while another might offer a bundle package. If both plans simplify to the same total cost equation based on your usage, you have infinitely many "break-even" points, meaning the cost is always the same regardless of which plan you choose. This system-of-equations logic is fundamental in economics, logistics, and engineering to optimize resources and understand underlying relationships between variables.