In a shop the cost of 2 pencils and 3 erasers is ₹9 and the cost of 4 pencils and 6 erasers is ₹18. Find the cost of each pencil and each eraser. [Equations: 2x+3y=9 (1) and 4x+6y=18 (2).]
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Step-by-Step Solution
Step 1: Define variables
Let x be the cost of one pencil and y be the cost of one eraser. This helps us set up our equations based on the given information.
Step 2: Formulate equations
According to the problem statement, the cost of 2 pencils and 3 erasers is ₹9, which gives us the first equation. Similarly, the cost of 4 pencils and 6 erasers is ₹18, leading to the second equation.
Step 3: Simplify the second equation
We can simplify the second equation by dividing all terms by 2. This reveals a relationship between the two equations that will help us solve the system. When divided by 2, 4x/2=2x, 6y/2=3y, and 18/2=9, resulting in 2x+3y=9.
Step 4: Compare equations
Upon comparing Equation (1) and Equation (3), we notice that they are identical. This means we essentially have only one unique linear equation for two variables, not a system of two independent equations.
Step 5: Express one variable in terms of the other
Since both equations are the same, we can express one variable in terms of the other. From 2x+3y=9, we can isolate x by first subtracting 3y from both sides, giving 2x=9−3y, and then dividing by 2 to get x=29−3y. This form allows us to find corresponding values of x for any chosen value of y.
Step 6: Conclusion about the solution
Since the two given equations are dependent (one is a multiple of the other), they represent the same line. Therefore, any pair of values (x, y) that satisfies one equation will also satisfy the other, leading to infinitely many possible solutions for the cost of each pencil and eraser. For example, if y=1, then x=29−3(1)=26=3. If y=3, then x=29−3(3)=20=0.