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Neokaal Study

Pair of Linear Equations in Two Variables

Competency Practice

Digit systems, parameter conditions for infinite/no solutions, and rates of motion.

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  1. The sum of the digits of a two-digit number is $9$. Also, $9$ times this number is twice the number obtained by reversing the order of the digits. Find the original number.

  2. Solve the pair of equations $5x - 4y = 7$ and $5x + 4y = 23$.

  3. Do the equations $5x + 4y = 12$ and $-15x - 12y = -36$ represent coincident lines? Justify your answer.

  4. Does the pair of equations $-2x - 3y = -3$ and $-2x - 3y = 1$ have no solution? Justify your answer.

  5. A rower moves at 8 km/h in still water. For the same 36 km distance, the upstream trip takes 3 times the downstream trip. Find the speed of the stream.

  6. A reading club charges one fixed amount for the first 2 days and then a daily extra amount. One reader paid Rs 18 for 6 days, and another paid Rs 14 for 4 days. Find the fixed charge and the extra charge per day.

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Answer Key

  1. $18$

    1. Let the tens digit be $x$ and the units digit be $y$. The original number is $10x + y$ and reversed number is $10y + x$.
    2. From the sum of digits: $x + y = 9$.
    3. From the second condition: $9(10x + y) = 2(10y + x) \implies 90x + 9y = 20y + 2x \implies 88x = 11y \implies y = 8x$.
    4. Substituting $y = 8x$ into $x + y = 9$ gives $9x = 9 \implies x = 1$ and $y = 8$.
    5. Therefore, the original number is $10(1) + 8 = 18$.

  2. The solution is $(3, 2)$.

    1. Eliminate one variable from $5x - 4y = 7$ and $5x + 4y = 23$.
    2. The determinant is $40$, so the pair has a unique solution.
    3. Solving gives x = $3$ and y = $2$.
    4. Therefore, the solution is $(3, 2)$.

  3. Yes. The pair is consistent and the lines are coincident.

    1. Compare the coefficient pairs of $5x + 4y = 12$ and $-15x - 12y = -36$.
    2. The determinant is $0$.
    3. All coefficients and constants are in the same ratio.
    4. So the lines are coincident and every point on the line is a solution.
    5. Therefore, the pair has infinitely many solutions.

  4. Yes. The pair is inconsistent and the lines are parallel.

    1. Compare the coefficient pairs of $-2x - 3y = -3$ and $-2x - 3y = 1$.
    2. The determinant is $0$.
    3. The x and y coefficients are proportional, but the constants are not in the same ratio.
    4. So the lines are parallel and the pair has no solution.
    5. Therefore, the pair has no solution.

  5. The stream speed is 4 km/h.

    1. Upstream speed is $8 - v$ and downstream speed is $8 + v$.
    2. The upstream time is 3 times the downstream time.
    3. So $\frac{36}{8 - v} = 3\times \frac{36}{8 + v}$.
    4. Solving gives v = $4$.

  6. The fixed charge is Rs 10 and the extra charge is Rs 2 per day.

    1. Let x be the fixed charge and y be the extra charge per day.
    2. The two bills give $x + 4y = 18$ and $x + 2y = 14$.
    3. Solving gives x = $10$ and y = $2$.