Khan Academy Practice Test 2 Math No Calculator Section Help

Test Preparation
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I found two questions difficult on the second Practice Test on Khan Academy and there were no explanations in the answer sections. Can anybody please help with these?

(Apologies if I’ve missed a mega thread for these kinds of posts, I skimmed the forum but didn’t see one).

First question:

Question 19 on the test.

ax + by = 12
2x + 8y = 60

In the systems of equations above, a and b are constants. If the system has infinitely many solutions, what is the value of a/b ?

Answer: 1/4

(I thought it would be 2/7)

Thanks in advance.

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Oops meant to post in SAT preparation.

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@AniseedLollies where did you get 2/7 from?

A system of two linear equations in two variables has infinitely many solutions if and only if they describe the same line. The only way for this to occur is if a = 2/5 and b = 8/5 (notice that multiplying both sides of the equation (2/5)x + (8/5)y = 12 produces the bottom equation). So a/b = 1/4.

For 20., one way you can do it is to drop an altitude from A to OB (call this point C, with coordinates (sqrt(3), 0). Then triangle AOC is a 30-60-90 triangle, with angle AOB = 30° (which is also π/6 radians).

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Thanks so much. For 19., I got to the a = 2/5 and b = 8/5 part but then incorrectly did 0.4/1.4 instead of 0.4/1.6 so that was really just a simple arithmetic error on my part that I am kicking myself for now! I couldn’t for the life of me understand where I had gone wrong so thanks for writing the answer, that’s what made it click.

I still don’t understand question 20 at all, do you know what part of the curriculum this is from so that I can read up on it?

Thank you so much for your prompt and helpful reply!

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@MITer94

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@AniseedLollies The main step in 20. is to recognize the 30-60-90 right triangle formed with hypotenuse AO. Also, knowing what radians are and how to convert from degrees to radians is useful.

If you know trig, another way is to see it is that angle AOB = arctan(1/sqrt(3)).

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@MITer94

Thanks so much for all your help. There is just one more thing, sorry about asking you all this but you’ve been really helpful.

One question for Level 3 Radical and rational equations (passport to advanced mathematics) is

What is the sum of the solutions to the above equation?
√4x+20 = x+2

(that square root symbol is meant to go over 4x+20)

I said 0 as +4 + -4 is zero but the answer said -4 is extraneous so the answer is just 4.

I don’t really understand this at all… why is -4 not valid?

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@AniseedLollies because if x = -4, then sqrt(4x+20) = sqrt(4) = 2, but x+2 = -2.

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@MITer94

Thank you so much.