What type of pair is formed by 16x+24y=88 and 2x+3y=11?
Answer and explanation
Correct answer: Consistent and dependent
Use the consistency criterion for two linear equations. The second equation is 2x+3y=11. Multiplying every term by 8 gives 16x+24y=88, which is precisely the first equation. Thus the coefficient ratios and constant ratio are equal: 16/2=24/3=88/11=8. Geometrically, both equations represent the same straight line, so they have infinitely many common points. Algebraically, there is only one independent equation for the two variables, leaving one variable free; this is why the pair is dependent. Because at least one common solution exists—in fact infinitely many—the pair is consistent as well. Therefore option A is the only correct answer. Inconsistent would mean no common solution, and independent would mean two non-proportional lines meeting at one point. Neither description fits these proportional equations.
Frequently asked questions
What is the correct answer to this question?
Consistent and dependent
Why is this the correct answer?
Use the consistency criterion for two linear equations. The second equation is 2x+3y=11. Multiplying every term by 8 gives 16x+24y=88, which is precisely the first equation. Thus the coefficient ratios and constant ratio are equal: 16/2=24/3=88/11=8. Geometrically, both equations represent the same straight line, so they have infinitely many common points. Algebraically, there is only one independent equation for the two variables, leaving one variable free; this is why the pair is dependent. Because at least one common solution exists—in fact infinitely many—the pair is consistent as well. Therefore option A is the only correct answer. Inconsistent would mean no common solution, and independent would mean two non-proportional lines meeting at one point. Neither description fits these proportional 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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