Question

Solve the following simultaneous equations:
$\frac{7}{2x+1}+\frac{13}{y+2}=27,\frac{13}{2x+1}+\frac{7}{y+2}=33$

Answer 1

We have, $\frac{7}{2x+1}+\frac{13}{y+2}=27...\left(i\right)$
and $\frac{13}{2x+1}+\frac{7}{y+2}=33....\left(ii\right)$
Substituting $\frac{1}{2x+1}=a$ and $\frac{1}{y+2}=b$ in equations $\left(i\right)$ and $\left(ii\right)$, we get
$7a+13b=27...\left(iii\right)$
and $13a+7b=33....\left(iv\right)$
Adding equations $\left(iii\right)$ and $\left(iv\right)$, we get
$20a+20b=60$
$a+b=3...\left(v\right)$
Subtracting equation $\left(iii\right)$ from equation $\left(iv\right)$, we get
$6a-6b=6$
$a-b=1...\left(vi\right)$
Adding equations $\left(v\right)$ and $\left(vi\right)$, we get
$2a=4\Rightarrow a=2$
Putting the value of $a$ in equation $\left(v\right)$, we get
$2+b=3\Rightarrow b=1$
Resubstituting the values of $a$ and $b$, we get
$\frac{1}{2x+1}=2$ and $\frac{1}{y+2}=1$
$1=4x+2$ and $1=y+2$
$x=-\frac{1}{4}$ and $y=-1$