To find the solution of the system of equations \( Y = -0.5x + 10 \) and \( Y = 0.75x \), we need to determine the point where the two lines intersect. This intersection point represents a solution to the system of equations, which can hold specific meanings depending on the context of the problem.
Step 1: Finding the Intersection Point
Set the two equations equal to each other:
\[ -0.5x + 10 = 0.75x \]
Now, solve for \( x \):
\[ 10 = 0.75x + 0.5x \] \[ 10 = 1.25x \] \[ x = \frac{10}{1.25} = 8 \]
Substituting \( x = 8 \) back into either of the equations to find \( y \):
Using \( Y = 0.75x \): \[ Y = 0.75(8) = 6 \]
Thus, the intersection point is \((8, 6)\).
Step 2: Understanding the Context
The interpretation of this solution depends on the real-world context from which these equations arise:
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If these equations represent the relationship between two variables, such as cost and revenue, profit and loss, or any two linear relationships:
- The point \((8, 6)\) could represent a situation where both relationships yield the same output (e.g., both cost and revenue allowing for a breakeven analysis).
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For example:
- Assume \( Y = -0.5x + 10 \) represents a cost function where \( x \) could represent the number of units produced, and \( Y \) represents the total cost.
- The equation \( Y = 0.75x \) could represent a revenue function for the same number of units produced.
- The solution \((8, 6)\) indicates that when 8 units are produced, the cost is $6, and the revenue is also $6. This suggests there is a breakeven point where costs and revenues are equal.
In any specific scenario, you would need to tailor the interpretation of the solution to the particulars of the equations based on what \( x \) and \( Y \) represent in that context.