If $\frac{a+1}{a^2-1} = \frac{1}{5}$, then the value of $a$ is -
Algebraic Equation Value for 'a'
We are given the equation: $ \frac{a+1}{a^2-1} = \frac{1}{5} $ Our objective is to determine the value of the variable $a$.
Equation Simplification Steps
First, simplify the denominator $a^2-1$. This is a difference of squares, which factors as:
$ a^2-1 = (a-1)(a+1) $Substitute this factorization back into the original equation:
$ \frac{a+1}{(a-1)(a+1)} = \frac{1}{5} $Solving for Variable 'a'
Provided that $a \neq -1$ (which means $a+1 \neq 0$), we can cancel the $(a+1)$ term from both the numerator and the denominator:
$ \frac{1}{a-1} = \frac{1}{5} $This simplified equation implies that the denominators must be equal:
$ a-1 = 5 $To find the value of $a$, add 1 to both sides of the equation:
$ a = 5 + 1 $ $ a = 6 $We must verify our assumption that $a \neq -1$. Since $a=6$, the condition $6 \neq -1$ holds true, confirming the validity of the cancellation step.
Thus, the calculated value of $a$ is 6.
Important Questions from Identities
- If $x + y + z = 10$, $x^2 + y^2 + z^2 = 38$, $5xyz = 150$ and $(xy + yz + zx) = 31$, then what is the value of $2(x^3 + y^3 + z^3)$?
- If $x = \sqrt[3]{31 + 14\sqrt{3}}$ and $y = \sqrt[3]{31 - 14\sqrt{3}}$, then the value of $x^3 + y^3 + 3x^3y^3$ is
- If $8x^3 - 27y^3 = (Ax + By)(Cx^2 - Dy^2 + Exy)$, then the value of $(A + B + C + D + E)$ is:
- A mother has decided to sum and product the ages of her children Ishan and Anushka. The sum and product of their ages are 18 and 56 respectively. The solution of which quadratic equation gives the possible ages of Ishan and Anushka?
- A mother has decided to sum and product the ages of her children Ishan and Anushka. The sum and product of their ages are 18 and 56, respectively. The solution of which quadratic equation gives the possible ages of Ishan and Anushka?
