Math CentralQuandaries & Queries


Question from Franco:

Hi There !

I am an intellectually very curious person with university degrees in the humanities field , but with a fairly poor mathematical background dating back to my High School days ( 50 years ago) , where I did Algebra, Trigonometry, etc.
I have now found a new interest in the CAUSAL aspect of reality , as opposed to the equally valid, but widely different ( although complementary )aspect dealt with in Philosophy and I ' d like to tackle Physics, with the ambition to gain some insight also into its more recent challenging developments, such as the Theory of of Relativity and Quantum Mechanics. I have read a lot about them in non-mathematical terms and I believe I have a basic grasp of their conceptual signification ( which for me, as in anything, is essential, even in Maths , at least as far as a mathematically unaided abstract thinking allows me !). I would like now to sharpen my mathematical tools, at least well enough to get me through some college level work. I am now honing my skills with the C.O.W. ( Calculus on the Web ) site, but I am am keeping my options open with other educational, sites, such as yours, which I have recently discovered. I have completed the pre-calculus level in a few months and I am now about to enter the « forbidden » land of Calculus, which I never did before. So far, by racking my brains, I have been able to get through, but in this case, I have hit a brick-wall and this is why I'd like to get your help.

I have read Professor Steve La Rocque's answer to a student regarding simultaneous equations . It is very well laid out, but I still have a problem regarding the following set of equations in relation to the equation of a circle.

We are investigating the circle passing through (3,1), (-2,0) and (-3,-4).
I have to substitute these points into the generic equation x^2+y^2+Dx+Ey=F

3 D + E - F = -10
-2 D - F = -4
-3 D - 4 E - F = -25

Although I have been able to solve correctly other drills on the same topic, on this one, with THREE equations, I have hit a brick-wall and I can't get out of the thicket even even by following Prof. Larocque's step-by-step explanation, Simply put, I cannot come up with the right values for D, E, F. Can somebody lay out the sequence with the solution , so that I can work my way backward and see where I was making the mistake.?

Thank you so much !



There are many ways to approach this problem but what I see is that the second equation only involves D and F so I would like another equation with only D and F as variables. To do so I multiply both sides of the first equation by 4 and then add the resulting equation to the third equation.

First equation times four:

12D + 4E - 4F = - 40

Third equation:

-3D - 4E - F = - 25


9D - 5F = - 65                (*)

Now I compare this equation to the second given equation and see that I can eliminate F by multiplying both sides of the second equation by 5.

Second equation times 5:

-10 D - 5F = -20

Equation (*):

9D - 5F = - 65


-19 D = 45
D = - 45/19

Substitute this value for D into equation (*) to find F. Finally substitute the values of D and F into any of the given equations and solve for E.

I hope this helps,

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