- System (a): Parallel lines with different y-intercepts, resulting in \(\boxed{\text{No solution}}\).
- System (b): The first equation leads to a contradiction (\(0=2\)), resulting in \(\boxed{\text{No solution}}\).
- System (c): A single linear equation with two variables results in \(\boxed{\text{Infinitely many solutions}}\).
- System (d): Substitution leads to a contradiction (\(0=-7\)), resulting in \(\boxed{\text{No solution}}\).
- System (e): Substitution yields the unique solution \(\boxed{\text{Exactly one solution: } (0, 0)}\).
Explanation
- Problem Analysis
We analyze five different systems of equations to determine the number of solutions and find the solutions where applicable. A system of linear equations in two variables, represented by \(x\) and \(y\), can have no solution if the lines are parallel and distinct, exactly one solution if they intersect at a single point, or infinitely many solutions if the equations represent the same line. For systems with one variable or single equations, we follow standard algebraic rules to determine consistency and dependency.
- System (a) Analysis
System (a) consists of the equations \(y = \frac{1}{2}x - 1\) and \(y = \frac{1}{2}x + 2\). Both equations describe lines with the same slope, \(m = \frac{1}{2}\), but different y-intercepts (\(-1\) and \(2\), respectively). Since the slopes are equal and the y-intercepts are different, the lines are parallel and never intersect. Therefore, there is no solution to this system.
- System (b) Analysis
System (b) is defined by \(x = x + 2\) and \(x = -3x - 2\). The first equation, \(x = x + 2\), simplifies by subtracting \(x\) from both sides to \(0 = 2\). This is a mathematical contradiction, meaning that no value of \(x\) can satisfy this equation. Because the system requires both equations to hold true simultaneously, the entire system has no solution.
- System (c) Analysis
System (c) consists of a single linear equation, \(y = 2x + 6\). A single equation in two variables defines a line in the coordinate plane. Since there are no further constraints or equations provided, any point \((x, y)\) that satisfies \(y = 2x + 6\) is a valid solution. For every value assigned to \(x\), there is a corresponding value for \(y\). Thus, there are infinitely many solutions.
- System (d) Analysis
System (d) consists of the equations \(y - 5x = -7\) and \(y = 5x\). We use the substitution method: substitute the expression for \(y\) from the second equation into the first. This gives \(5x - 5x = -7\), which simplifies to \(0 = -7\). This is a contradiction, which indicates that the lines are parallel and distinct. Therefore, there is no solution to this system.
- System (e) Analysis and Conclusion
System (e) consists of the equations \(y = 20x\) and \(20y = x\). We use substitution: substitute \(y = 20x\) into the second equation: \(20(20x) = x\), which simplifies to \(400x = x\). Subtracting \(x\) from both sides gives \(399x = 0\), so \(x = 0\). Substituting \(x = 0\) back into the first equation, we find \(y = 20(0) = 0\). Thus, the only solution to this system is the point \((0, 0)\), which means there is exactly one solution.
Examples
Understanding how to analyze systems of linear equations is fundamental in various fields, such as engineering and economics. For instance, in economics, a system might represent the supply and demand for a product; the point of intersection represents the market equilibrium, where the quantity supplied equals the quantity demanded at a specific price. If the equations are parallel, it might indicate that there is no price point where supply and demand can balance, highlighting a market disequilibrium. Mastering these techniques helps in modeling real-world situations effectively.