What is found by comparing the ratios of (a) and (b) in (9x+4y=21) and (4x+2y=10)?
Answer and explanation
Correct answer: (9/4 \ne 4/2), so one unique solution
For a pair of linear equations, if the ratios of the coefficients of x and y are unequal, the two equations represent lines with different slopes. Such lines intersect at exactly one point. Consequently, the system is consistent and has one unique solution. Infinite solutions would require both variable-coefficient ratios and the constant ratio to agree, while no solution would require equal variable ratios but a different constant ratio.
Here, \\(9/4=2.25\\), whereas \\(4/2=2\\). Therefore \\(9/4\\ne4/2\\), so the lines are not parallel and cannot be the same line. They meet once, giving a unique ordered pair \\(x,y\\). Thus option C is the correct conclusion from the comparison.
Frequently asked questions
What is the correct answer to this question?
(9/4 \ne 4/2), so one unique solution
Why is this the correct answer?
For a pair of linear equations, if the ratios of the coefficients of x and y are unequal, the two equations represent lines with different slopes. Such lines intersect at exactly one point. Consequently, the system is consistent and has one unique solution. Infinite solutions would require both variable-coefficient ratios and the constant ratio to agree, while no solution would require equal variable ratios but a different constant ratio.
Here, \\(9/4=2.25\\), whereas \\(4/2=2\\). Therefore \\(9/4\\ne4/2\\), so the lines are not parallel and cannot be the same line. They meet once, giving a unique ordered pair \\(x,y\\). Thus option C is the correct conclusion from the comparison.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Pair of Linear Equations in Two Variables. Topic: Conditions for solvability.
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