Solving Cost Problems Using Linear Equations

This problem involves finding the cost of one mobile phone given two scenarios involving the combined cost of LED TVs and mobile phones. We can solve this by setting up a system of linear equations.

Setting Up the Equations for Cost

Let's define variables for the unknown costs:

• Let \(L\) represent the cost of one LED TV in ₹.
• Let \(M\) represent the cost of one mobile phone in ₹.

Based on the information given in the question, we can form two linear equations:

• The first statement says "two LED TVs and one mobile phone cost ₹31,000". This translates to:

\begin{equation} 2L + M = 31000 \quad (Equation \, 1) \end{equation}

• The second statement says "two mobile phones and one LED TV cost ₹35,000". This translates to:

\begin{equation} L + 2M = 35000 \quad (Equation \, 2) \end{equation}

Solving the System of Equations

We now have a system of two linear equations with two variables (\(L\) and \(M\)). We want to find the value of \(M\), the cost of one mobile phone.

We can use the elimination method or the substitution method. Let's use the elimination method.

Our goal is to eliminate one variable so we can solve for the other. We can eliminate \(L\) by making the coefficient of \(L\) the same in both equations.

• Multiply Equation 2 by 2:

\begin{align*} 2 \times (L + 2M) &= 2 \times 35000 \\ 2L + 4M &= 70000 \quad (Equation \, 3) \end{align*}

• Now we have Equation 1 ($2L + M = 31000$) and Equation 3 ($2L + 4M = 70000$). The coefficient of \(L\) is the same in both equations.
• Subtract Equation 1 from Equation 3:

\begin{align*} (2L + 4M) - (2L + M) &= 70000 - 31000 \\ 2L + 4M - 2L - M &= 39000 \\ (2L - 2L) + (4M - M) &= 39000 \\ 0 + 3M &= 39000 \\ 3M &= 39000 \end{align*}

• Now, solve for \(M\):

\begin{align*} M &= \frac{39000}{3} \\ M &= 13000 \end{align*}

So, the value of one mobile phone is ₹13,000.

Verification (Optional)

We can find the value of \(L\) and check if the original equations hold true.

• Substitute \(M = 13000\) into Equation 1:

\begin{align*} 2L + 13000 &= 31000 \\ 2L &= 31000 - 13000 \\ 2L &= 18000 \\ L &= \frac{18000}{2} \\ L &= 9000 \end{align*}

The cost of one LED TV is ₹9,000.

• Now, check if these values satisfy Equation 2:

\(L + 2M = 9000 + 2(13000) = 9000 + 26000 = 35000\)

Since \(35000 = 35000\), the values \(L = 9000\) and \(M = 13000\) are correct.

Summary of Costs

The value of one mobile phone is ₹13,000.

Revision Table: Linear Equations in Cost Problems

Additional Information: Solving Word Problems

Solving word problems like this one often follows a structured approach:

• Read Carefully: Understand the problem and identify what is being asked.
• Define Variables: Assign letters to the unknown quantities you need to find.
• Set Up Equations: Translate the words into mathematical equations based on the relationships described.
• Solve the System: Use algebraic methods (substitution or elimination) to find the values of the variables.
• Check Your Answer: Substitute the values you found back into the original equations to ensure they are correct.
• State the Answer: Clearly answer the question asked in the problem in the context of the word problem.

This problem demonstrates how systems of linear equations are useful tools for solving real-world problems involving costs and quantities.