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Linear Equations: Competency Practice

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

6 problems·20–25 min·★★★★☆
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  1. Problem 1
    linear-equations·★★★★★
    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.
    ▶Answer
    $18$
    ▶Step-by-step solution
    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 $10(1) + 8 = 18$.
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  2. Problem 2
    Solving Linear Pairs·★★★☆☆
    Solve the pair of equations $5x - 4y = 7$ and $5x + 4y = 23$.
    ▶Answer
    The solution is $(3, 2)$.
    ▶Step-by-step solution
    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 $(3, 2)$.
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  3. Problem 3
    Consistency Reasoning·★★★★☆
    Do the equations $5x + 4y = 12$ and $-15x - 12y = -36$ represent coincident lines? Justify your answer.
    ▶Answer
    Yes. The pair is consistent and the lines are coincident.
    ▶Step-by-step solution
    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.
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  4. Problem 4
    Consistency Reasoning·★★★★☆
    Does the pair of equations $-2x - 3y = -3$ and $-2x - 3y = 1$ have no solution? Justify your answer.
    ▶Answer
    Yes. The pair is inconsistent and the lines are parallel.
    ▶Step-by-step solution
    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.
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  5. Problem 5
    Word Problems·★★★★★
    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.
    ▶Answer
    The stream speed is 4 km/h.
    ▶Step-by-step solution
    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 = $4$.
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  6. Problem 6
    Word Problems·★★★★☆
    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.
    ▶Answer
    The fixed charge is Rs 10 and the extra charge is Rs 2 per day.
    ▶Step-by-step solution
    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 = $2$.
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