\(\sqrt{x^2+y^2}=100 \)
\(\sqrt{\left(x+\frac{4800}{\sqrt{937}}\right)^2+\left(y+\frac{3800}{\sqrt{937}}\right)^2}=299.289496902\)
Tried to write it in \(\LaTeX\) to be easier on the eyes.
+2441 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:
\(x≈68.53740254003346, y≈72.81912147963209\)
\(x≈86.60254037943613, y≈49.99999999828134\)
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
