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# Messy Problem

+1
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$$\sqrt{x^2+y^2}=100$$

$$\sqrt{\left(x+\frac{4800}{\sqrt{937}}\right)^2+\left(y+\frac{3800}{\sqrt{937}}\right)^2}=299.289496902$$

Tried to write it in $$\LaTeX$$ to be easier on the eyes.

Oct 8, 2017

#1
+2340
+1

This is indeed a messy problem. I'll solve for both variables, I guess.

1. Solve for a Variable

In this case, I will solve for x in the first equation, $$\sqrt{x^2+y^2}=100$$

 $$\sqrt{x^2+y^2}=100$$ The first step is to square both sides so that we eliminate the square root symbol. $$x^2+y^2=10000$$ Subtract $$y^2$$ from both sides. $$x^2=10000-y^2$$ Take the square root from both sides to isolate x. $$x=\sqrt{10000-y^2}$$

2. Use Substitution to eliminate the x and solve for y

Plug in $$\sqrt{10000-y^2}$$ for x into the 2nd equation and solve for y. This is not going to look good...

 $$\sqrt{\left(x+\frac{4800}{\sqrt{937}}\right)^2+\left(y+\frac{3800}{\sqrt{937}}\right)^2}=299.289496902$$ Plug in the value for x. $$\sqrt{\left(\sqrt{10000-y^2}+\frac{4800}{\sqrt{937}}\right)^2+\left(y+\frac{3800}{\sqrt{937}}\right)^2}=299.289496902$$ Square both sides to eliminate the square root. $$\textcolor{blue}{\left(\sqrt{10000-y^2}+\frac{4800}{\sqrt{937}}\right)^2}+\left(y+\frac{3800}{\sqrt{937}}\right)^2=299.289496902^2$$ In both binomials, we must follow the rule that $$(a+b)^2=a^2+2ab+b^2$$. I'll do the first binomial first, in blue. $$\left(\sqrt{10000-y^2}+\frac{4800}{\sqrt{937}}\right)^2=\left(\sqrt{10000-y^2}\right)^2+2\sqrt{10000-y^2}*\frac{4800}{\sqrt{937}}+\left(\frac{4800}{\sqrt{937}}\right)^2$$ Simplify this. Here we go... $$10000-y^2+\frac{9600\sqrt{10000-y^2}}{\sqrt{937}}+\frac{23040000}{937}$$ Now, let's remember that this equals the bit in blue. Now, let's expand the other part, in normal text.This time, I'll expand it without showing much.

 $$\left(y+\frac{3800}{\sqrt{937}}\right)^2$$ Expand this using the same technique as above. $$y^2+\left(2y*\frac{3800}{\sqrt{937}}\right)+\left(\frac{3800}{\sqrt{937}}\right)^2$$ Simplify further. $$y^2+\frac{7600y}{\sqrt{937}}+\frac{14440000}{937}$$

Time to do the simplification process.

 $$10000-y^2+\frac{9600\sqrt{10000-y^2}}{\sqrt{937}}+\frac{23040000}{937}+y^2+\frac{7600y}{\sqrt{937}}+\frac{14440000}{937}$$ To make this easier to digest, I will rearrange the terms. $$y^2-y^2+\frac{9600\sqrt{10000-y^2}}{\sqrt{937}}+\frac{7600y}{\sqrt{937}}+10000+\frac{23040000}{937}+\frac{14440000}{937}$$ Now, let's do the simplification process. $$\frac{9600\sqrt{10000-y^2}+7600y}{\sqrt{937}}+50000$$ Great! Now that we have simplified as much as possible, let's reinsert this into the equation.

 $$\frac{9600\sqrt{10000-y^2}+7600y}{\sqrt{937}}+50000=299.289496902^2$$ We have to get rid of the remaining fraction by multiplying by the square root of 937 on all sides. $$9600\sqrt{10000-y^2}+7600y+50000\sqrt{937}=299.289496902^2\sqrt{937}$$ Subtract $$7600y+50000\sqrt{937}$$ $$9600\sqrt{10000-y^2}=229.289496902^2\sqrt{937}-7600y-50000\sqrt{937})$$ Doing some more simplifying...

You know what...

This is really boring and tedious...

There are 2 ordered pair solutions to this. They are the following:

$$x≈68.53740254003346, y≈72.81912147963209$$

$$x≈86.60254037943613, y≈49.99999999828134$$

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Oct 8, 2017
#2
+1

Sqrt(x^2 + y^2) =100

x =sqrt(10,000 - y^2)     Sub this into the second equation:

Solve for y:
sqrt((y + 3800/sqrt(937))^2 + (sqrt(10000 - y^2) + 4800/sqrt(937))^2) = 299.289

Raise both sides to the power of two:
(y + 3800/sqrt(937))^2 + (sqrt(10000 - y^2) + 4800/sqrt(937))^2 = 89574.2

Rewrite the left hand side by combining fractions. (y + 3800/sqrt(937))^2 + (sqrt(10000 - y^2) + 4800/sqrt(937))^2 = 400/937 (117125 + 19 sqrt(937) y + 24 sqrt(937) sqrt(10000 - y^2)):
400/937 (117125 + 19 sqrt(937) y + 24 sqrt(937) sqrt(10000 - y^2)) = 89574.2

Multiply both sides by 937/400:
117125 + 19 sqrt(937) y + 24 sqrt(937) sqrt(10000 - y^2) = 209828.

Subtract 19 sqrt(937) y + 117125 from both sides:
24 sqrt(937) sqrt(10000 - y^2) = 92702.6 - 19 sqrt(937) y

Raise both sides to the power of two:
539712 (10000 - y^2) = (92702.6 - 19 sqrt(937) y)^2

Expand out terms of the left hand side:
5397120000 - 539712 y^2 = (92702.6 - 19 sqrt(937) y)^2

Expand out terms of the right hand side:
5397120000 - 539712 y^2 = 338257 y^2 - 1.07831×10^8 y + 8.59377×10^9

Subtract 338257 y^2 - 1.07831×10^8 y + 8.59377×10^9 from both sides:
-877969 y^2 + 1.07831×10^8 y - 3.19665×10^9 = 0

Divide both sides by -877969:
y^2 - 122.819 y + 3640.96 = 0

Subtract 3640.96 from both sides:
y^2 - 122.819 y = -3640.96

Add 3771.13 to both sides:
y^2 - 122.819 y + 3771.13 = 130.178

Write the left hand side as a square:
(y - 61.4096)^2 = 130.178

Take the square root of both sides:
y - 61.4096 = 11.4096 or y - 61.4096 = -11.4096

Add 61.4096 to both sides:
y = 72.8191 or y - 61.4096 = -11.4096

Add 61.4096 to both sides:
y = 72.8191        or         y = 50.      or         x=68.5374     and           x=86.60254

Oct 9, 2017