Explanation for Question 21 From the Math (Calc) Section on the 2020 October Sat
So question number 21 is asking us how many solutions a pair of lines 2 has. And so for Eliza has zero solutions. 3 They have to be parallel and for lines to be parallel, 4 they have to have the same slope. Now, 5 if we look at our two equations, they're not written in y-intercept form, 6 but we can easily convert them into Y intercept form by moving the Y 7 and the Y intercepts around. 8 And so that gives us this form of the equations right here. 9 Now, if we look at the slopes, right, 10 our slopes are different. It's two versus four. So that means they can't be 11 parallel. So a is no good. 12 If we look at infinitely many infinitely, 13 many means that they're both the exact same line. 14 They both lie and have the same point. 15 So that would be like one really thick line on a, 16 on a coordinate plane. 17 And it says these both have different slopes. Again, they can't be the same 18 line, right? Cause the same line would have the same slope and the same 19 y-intercept so this is no good. 20 Now to have two solutions, the line kind of has to have a curve 21 to it, right? Because we think if I put a line here, 22 the only way for something to hit it twice is for it to curve 23 back around to hit it the second time. 24 And since these are both lines, it's impossible to curve, 25 right? So I can't have two solutions. I can only have one solution and 26 you can also do that by just graphing it. Right? 27 Both of them have a y-intercept of negative three, 28 one increases with a slope of two and then one increases with a steeper 29 slope of four. But the only point where they're going to intersect is on 30 that Y axis. So our correct answer is B.