How many solutions does (4x+5y=1) and (8x+9y=2) have?
Answer and explanation
Correct answer: Exactly one solution
Compare the ratios of the coefficients: \(\frac{a_1}{a_2}=\frac{4}{8}=\frac{1}{2}\), whereas \(\frac{b_1}{b_2}=\frac{5}{9}\). Since \(\frac{a_1}{a_2}\ne\frac{b_1}{b_2}\), the two lines intersect at exactly one point, so the pair has a unique solution. Infinitely many solutions would require all three ratios to be equal. Exam tip: compare \(a_1/a_2\) and \(b_1/b_2\) first.
Frequently asked questions
What is the correct answer to this question?
Exactly one solution
Why is this the correct answer?
Compare the ratios of the coefficients: \(\frac{a_1}{a_2}=\frac{4}{8}=\frac{1}{2}\), whereas \(\frac{b_1}{b_2}=\frac{5}{9}\). Since \(\frac{a_1}{a_2}\ne\frac{b_1}{b_2}\), the two lines intersect at exactly one point, so the pair has a unique solution. Infinitely many solutions would require all three ratios to be equal. Exam tip: compare \(a_1/a_2\) and \(b_1/b_2\) first.
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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