The pair of equations $ 3x - 5y = 7 $ and $-6x + 10y = 7 $ have: | Mathematics MCQ | Examoogle

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The pair of equations $3x - 5y = 7$ and $-6x + 10y = 7$ have:

a unique solution

infinitely many solutions

no solution

two solutions

Correct Answer

📐 MA • Math Coefficient ComparisonMathematicsAlgebra

Option C

no solution

Quick Summary:

Observe the ratio of coefficients: 3/-6 = -5/10 = -1/2, but 7/7 = 1. Since the x and y ratios are equal but different from the constant term ratio, the lines are parallel and have no intersection.

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📐MAMath SolutionCoefficient Comparison

📋 Given

Two linear equations in two variables: 3x - 5y = 7 and -6x + 10y = 7

🔢 Formula Used

a1a2=b1b2≠c1c2\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}a2​a1​​=b2​b1​​=c2​c1​​

⚡ Exam Hall Shortcut / Speed Trick

Observe the ratio of coefficients: 3/-6 = -5/10 = -1/2, but 7/7 = 1. Since the x and y ratios are equal but different from the constant term ratio, the lines are parallel and have no intersection.

⚠️ Common Student Trap / Pitfall

Many students rush to assume that if the ratios of x and y coefficients are identical, the system must have infinitely many solutions, forgetting to check the constant term.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Identify coefficients

Given the system in standard form $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$ , we have: $a_1 = 3, b_1 = -5, c_1 = 7$ and $a_2 = -6, b_2 = 10, c_2 = 7$ .

a1=3,b1=−5,c1=7,a2=−6,b2=10,c2=7a_1=3, b_1=-5, c_1=7, a_2=-6, b_2=10, c_2=7a1​=3,b1​=−5,c1​=7,a2​=−6,b2​=10,c2​=7

Check ratios

Calculate the ratios of the coefficients: $a_1/a_2 = 3/-6 = -1/2$ , $b_1/b_2 = -5/10 = -1/2$ , and $c_1/c_2 = 7/7 = 1$ .

3−6=−510≠77\frac{3}{-6} = \frac{-5}{10} \neq \frac{7}{7}−63​=10−5​=77​

Conclusion

Since $a_1/a_2 = b_1/b_2 \neq c_1/c_2$ , the lines represented by these equations are parallel and non-coincident. Thus, the system has no solution.

NosolutionNo solutionNosolution

✅

C is correct because the ratios of the coefficients of x and y are equal, but they are not equal to the ratio of the constant terms, indicating parallel lines with no intersection.

Core Concepts Used

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Linear equations System consistency Parallel lines

💡 EXAM TIP

This condition ( $a_1/a_2 = b_1/b_2 \neq c_1/c_2$ ) is a fundamental check in coordinate geometry for determining if two lines are parallel.

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