- Identify the slope-intercept form for each equation: \(y = 2x - 5\) and \(y = -2x + 5\).
- Compare the slopes (\(2
eq -2\)) to conclude that the lines are not parallel and therefore intersect at exactly one point.
- Solve the resulting equation \(2x - 5 = -2x + 5\) to find the x-coordinate \(x = 2.5\).
- Determine the y-coordinate \(y = 0\) by substitution, confirming the solution \((2.5, 0)\).
- Final answer: \(\boxed{\text{one solution: (2.5, 0)}}\).
Explanation
- Analysis of the linear system
We are given a system of two linear equations in two variables, \(x\) and \(y\):
- \(y = 2x - 5\)
- \(-8x - 4y = -20\)
The goal is to determine the number of solutions for this system. A linear system can have one solution if the lines intersect, no solutions if the lines are parallel and distinct, or an infinite number of solutions if the lines are coincident. We can determine this by comparing their slopes and y-intercepts or by solving for the point of intersection.
- Comparing the two equations
To solve this system, we can express both equations in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
The first equation is already in this form:
\[y = 2x - 5\]
Here, the slope is \(m_1 = 2\) and the y-intercept is \(b_1 = -5\).
Now, let's convert the second equation, \(-8x - 4y = -20\), into slope-intercept form:
First, add \(8x\) to both sides:
\[-4y = 8x - 20\]
Next, divide every term by \(-4\):
\[y = \frac{8x}{-4} - \frac{20}{-4}\]
\[y = -2x + 5\]
Here, the slope is \(m_2 = -2\) and the y-intercept is \(b_2 = 5\).
- Finding the point of intersection
By comparing the two equations, we see that the slopes (\(2\) and \(-2\)) are different. Since the slopes are different, the lines are not parallel, which means they must intersect at exactly one point.
To find this intersection point, we set the two expressions for \(y\) equal to each other:
\[2x - 5 = -2x + 5\]
Add \(2x\) to both sides:
\[4x - 5 = 5\]
Add \(5\) to both sides:
\[4x = 10\]
Divide by \(4\):
\[x = 2.5\]
Now substitute \(x = 2.5\) back into the first equation to find \(y\):
\[y = 2(2.5) - 5\]
\[y = 5 - 5\]
\[y = 0\]
The point of intersection is \((2.5, 0)\).
- Conclusion
The system of equations has exactly one solution, which is the intersection point \((2.5, 0)\). Comparing this result with the given options, we select "one solution: \((2.5, 0)\)".
Examples
Imagine you are planning a small business. One equation might represent your production costs, where \(y\) is the cost and \(x\) is the number of units produced. The second equation could represent your total revenue based on sales. By finding the point where these two lines intersect, you can determine your "break-even point"—the exact number of units you need to produce and sell to cover your costs without making a profit or a loss. This simple algebraic exercise is a fundamental tool for real-world decision-making.