What is found by comparing the ratios of (a) and (b) in (7x+3y=19) and (2x+y=6)?
Answer and explanation
Correct answer: (7/2 \ne 3/1), so one unique solution
For equations written as \\(a_1x+b_1y+c_1=0\\) and \\(a_2x+b_2y+c_2=0\\), unequal ratios \\(a_1/a_2\\) and \\(b_1/b_2\\) mean that the two lines have different slopes. Lines with different slopes meet at exactly one point, so the pair has a unique solution. Equality of the first two ratios is not required here.
In this pair, \\(7/2=3.5\\), while \\(3/1=3\\). Hence \\(7/2\\ne3/1\\). The lines therefore have different directions and intersect once. The constant ratio is not needed to establish uniqueness after the first two ratios are found unequal. Thus option C correctly states that there is one unique solution.
Frequently asked questions
What is the correct answer to this question?
(7/2 \ne 3/1), so one unique solution
Why is this the correct answer?
For equations written as \\(a_1x+b_1y+c_1=0\\) and \\(a_2x+b_2y+c_2=0\\), unequal ratios \\(a_1/a_2\\) and \\(b_1/b_2\\) mean that the two lines have different slopes. Lines with different slopes meet at exactly one point, so the pair has a unique solution. Equality of the first two ratios is not required here.
In this pair, \\(7/2=3.5\\), while \\(3/1=3\\). Hence \\(7/2\\ne3/1\\). The lines therefore have different directions and intersect once. The constant ratio is not needed to establish uniqueness after the first two ratios are found unequal. Thus option C correctly states that there is one unique solution.
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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