What type of pair is formed by 18x+30y=126 and 3x+5y=21?
Answer and explanation
Correct answer: Consistent and dependent
The governing concept is identification of coincident equations using proportional coefficients. Multiply the second equation, 3x+5y=21, by 6. We obtain 18x+30y=126, exactly the first equation. Therefore 18/3=30/5=126/21=6. Equal ratios for both variable coefficients and the constants show that the two equations have identical standard form after scaling. Their graphs are the same line, not parallel separate lines, so every point on that line is a common solution. Hence the pair has infinitely many solutions and is consistent and dependent. Option A is correct. An inconsistent pair would have equal ratios for the variable coefficients but a different ratio for the constants. A consistent independent pair would have non-proportional coefficients and one unique intersection. Neither alternative describes the given equations.
Frequently asked questions
What is the correct answer to this question?
Consistent and dependent
Why is this the correct answer?
The governing concept is identification of coincident equations using proportional coefficients. Multiply the second equation, 3x+5y=21, by 6. We obtain 18x+30y=126, exactly the first equation. Therefore 18/3=30/5=126/21=6. Equal ratios for both variable coefficients and the constants show that the two equations have identical standard form after scaling. Their graphs are the same line, not parallel separate lines, so every point on that line is a common solution. Hence the pair has infinitely many solutions and is consistent and dependent. Option A is correct. An inconsistent pair would have equal ratios for the variable coefficients but a different ratio for the constants. A consistent independent pair would have non-proportional coefficients and one unique intersection. Neither alternative describes the given equations.
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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