- Express the second equation in slope-intercept form to identify the slopes: \(y = 2x - 5\) and \(y = -2x + 5\).
- Compare the slopes to confirm they are different, implying a single intersection point.
- Solve the system using substitution, setting \(2x - 5 = -2x + 5\) to find the coordinate \(x = 2.5\).
- Calculate \(y = 0\) to confirm the intersection point \((2.5, 0)\).
- Final Answer: \(\boxed{\text{one solution: } (2.5, 0)}\)
Explanation
- Problem Analysis
To determine the number of solutions for the given system of linear equations, we first analyze the structure of the equations:
1) \(y = 2x - 5\)
2) \(-8x - 4y = -20\)
A system of two linear equations in two variables can either have one unique solution, no solutions (if the lines are parallel and distinct), or infinitely many solutions (if the lines are identical). We will test these possibilities by examining the slope-intercept form (\(y = mx + b\)) or using substitution.
- Slope and Intercept Comparison
We aim to find the solution set for the system. Let's convert the second equation into the slope-intercept form \(y = mx + b\) to compare it with the first:
Starting with \(-8x - 4y = -20\):
Subtract \(-8x\) from both sides: \(-4y = 8x - 20\).
Divide both sides by \(-4\): \(y = \frac{8x}{-4} - \frac{20}{-4}\), which simplifies to \(y = -2x + 5\).
Now, compare the two equations:
1) \(y = 2x - 5\)
2) \(y = -2x + 5\)
The slopes are \(2\) and \(-2\), respectively. Since the slopes are different, the lines are not parallel and must intersect at exactly one point.
- Solving the System
Since the lines have different slopes, there is exactly one solution. To find it, we substitute the expression for \(y\) from the first equation into the second equation:
\(-8x - 4(2x - 5) = -20\)
Distribute the \(-4\):
\(-8x - 8x + 20 = -20\)
Combine like terms:
\(-16x + 20 = -20\)
Subtract 20 from both sides:
\(-16x = -40\)
Divide by \(-16\):
\(x = \frac{-40}{-16} = 2.5\)
Now, substitute \(x = 2.5\) back into the first equation to find \(y\):
\(y = 2(2.5) - 5 = 5 - 5 = 0\)
The solution is \((2.5, 0)\).
- Conclusion
The system of linear equations has exactly one solution at the point \((2.5, 0)\). This matches the second option provided. Thus, the correct identification is one solution: \((2.5, 0)\). The final answer is \(\boxed{\text{one solution: } (2.5, 0)}\).
Examples
This math problem is essential for understanding how to find intersection points, which has practical applications in fields like economics. For example, if a company's production costs are represented by one linear equation and their revenue by another, the point where these two lines intersect—the break-even point—is found using exactly the same method of solving a system of linear equations. If \(y = 2x - 5\) represents revenue and \(y = -2x + 5\) represents costs, the intersection at \(x = 2.5\) tells the company exactly how many units they need to sell to reach zero profit, which is a vital metric for business planning.