MP Board · Class 10 · Mathematics · Pair of Linear Equations in Two VariablesRitu can row downstream $20\text{ km}$ in $2\text{ hours}$, and upstream $4\text{ km}$ in $2\text{ hours}$. Formulate the system of linear equations representing this situation. Find her speed of rowing in still water and the speed of the current using the elimination method. Show complete step-by-step calculations and verification.

Step 1: Define variables\nLet the speed of Ritu in still water = $x\text{ km/h}$\nLet the speed of the stream (current) = $y\text{ km/h}$

Step 2: Determine relative speeds

• Speed downstream $= (x + y)\text{ km/h}$
• Speed upstream $= (x - y)\text{ km/h}$ \nWe know the formula: $$\text{Speed} = \frac{\text{Distance}}{\text{Time}} \quad \implies \quad \text{Distance} = \text{Speed} \times \text{Time}$$

Step 3: Formulate Linear Equations

Condition 1: Downstream motion

• Distance = $20\text{ km}$, Time = $2\text{ hours}$ $$(x + y) \times 2 = 20$$ $$x + y = 10 \quad \text{--- (Equation 1)}$$

Condition 2: Upstream motion

• Distance = $4\text{ km}$, Time = $2\text{ hours}$ $$(x - y) \times 2 = 4$$ $$x - y = 2 \quad \text{--- (Equation 2)}$$

Step 4: Solve the system using Elimination Method

\nAdding Equation (1) and Equation (2): $$(x + y) + (x - y) = 10 + 2$$ $$2x = 12$$ $$x = \frac{12}{2} = 6$$ \nSubstitute $x = 6$ into Equation (1): $$6 + y = 10$$ $$y = 10 - 6 = 4$$

Step 5: Verification of Results

• Downstream speed $= 6 + 4 = 10\text{ km/h}$ Time taken $= \frac{20}{10} = 2\text{ hours}$ (Matches given data)
• Upstream speed $= 6 - 4 = 2\text{ km/h}$ Time taken $= \frac{4}{2} = 2\text{ hours}$ (Matches given data)

Final Answer

• Speed of rowing in still water $= 6\text{ km/h}$
• Speed of the current $= 4\text{ km/h}$