<s>[INST] Little help with some algebra Note: This isn't homework, I'm skipping ahead of class. Please answer all these equations, I'm deathly stuck on them.
Use the substitution method only please. (Find $x$ and $y$.)
\begin{align}
  ax\left({\frac {1}{a-b}-\frac {1}{a+b}}\right)+by\left({\frac {1}{b-a}-\frac {1}{b+a}}\right)=2 
 \end{align}
$a$ is not equal to $b$ or $-b$
Note - It's $ax$ as the first word, I am worried the latex might mess up there.
Equation 2:
\begin{align}
 6x+5y=7x+3y+1=2\left({x+6y-1}\right)
\end{align}
Equation 3:
\begin{align}
 \sqrt{2}x+\sqrt{3}y=0
\end{align}
\begin{align}
 \sqrt{3}x-\sqrt{8}y=0
\end{align}
Thank you for the help!
Note: Please provide the full solution, the more hints i get the more confused i get...
 [/INST] First Equation:
$$ax\left({\frac {1}{a-b}-\frac {1}{a+b}}\right)+by\left({\frac {1}{b-a}-\frac {1}{b+a}}\right)=2 $$
$$ax\left({\frac {a+b-a+b}{a^2-b^2}}\right)+by\left({\frac {b+a-b+a}{b^2-a^2}}\right)=2$$
$$2abx-2aby=2(a^2-b^2)$$
$$x-y=\frac{a}{b}-\frac{b}{a}$$
From this the obvious solution is $x=a/b \text{ and } y=b/a$. But there will be infinite solutions $x=y+(a/b)-(b/a)$. Just put values for $y$ and get corresponding values for $x$.
Second Equation:
$$6x+5y=7x+3y+1=2(x+6y-1)$$
Break it into $2$ equations
$$6x+5y=7x+3y+1 \text{ and } 6x+5y=2(x+6y-1)$$
$$-x+2y=1 \text{ and } 4x-7y=-2$$
The value of $x$ from the first equation is $x=2y-1$. Put it in second equation.
$$4(2y-1)-7y=-2$$
$$y=2$$
Hence $x=2y-1=3$.
Third equation:
$$\sqrt2x+\sqrt3y=0 $$
From this $x=-\sqrt{3/2}y$. Put this in the second equation.
$$\sqrt3x-\sqrt8y=0$$
$$y(3/\sqrt2 - \sqrt8)=0$$
$$y=0$$
Hence $x=0$.
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