Algebraic and Geometrical Representation of Cost Problem
Video Explanation
Question
The cost of 2 kg of apples and 1 kg of grapes on a day was ₹160. After a month, the cost of 4 kg of apples and 2 kg of grapes is ₹300. Represent the situation algebraically and geometrically.
Solution
Step 1: Define the Variables
Let
\[ x = \text{cost of 1 kg of apples (₹)} \]
\[ y = \text{cost of 1 kg of grapes (₹)} \]
Step 2: Form the Linear Equations
First Condition
Cost of 2 kg apples and 1 kg grapes = ₹160
\[ 2x + y = 160 \quad \text{(Equation 1)} \]
Second Condition
Cost of 4 kg apples and 2 kg grapes = ₹300
\[ 4x + 2y = 300 \quad \text{(Equation 2)} \]
Step 3: Algebraic Representation
The algebraic representation of the situation is the pair of linear equations:
\[ \boxed{2x + y = 160} \]
\[ \boxed{4x + 2y = 300} \]
Step 4: Geometrical Representation
Equation (1): \(2x + y = 160\)
\[ y = 160 – 2x \]
| x | y |
|---|---|
| 0 | 160 |
| 80 | 0 |
Equation (2): \(4x + 2y = 300\)
\[ 2x + y = 150 \Rightarrow y = 150 – 2x \]
| x | y |
|---|---|
| 0 | 150 |
| 75 | 0 |
Step 5: Graphical Interpretation
Plot the points:
- For Equation (1): (0,160) and (80,0)
- For Equation (2): (0,150) and (75,0)
Join each pair of points to obtain two straight lines.
The two lines are parallel and do not intersect.
Conclusion
The given situation is represented by the equations:
\[ 2x + y = 160 \quad \text{and} \quad 4x + 2y = 300 \]
Since the graphs are parallel, the system has no solution.
Hence, the given data is inconsistent.