The equation (24 x^2+25 x-47)/(a x-2)=-8 x-3-(53)/(a x-2) is true for all values of x ≠(2)/(a),

step 1 Okay, so if we're looking for the value of A, let's first go ahead and try to simplify this comp

The equation (24 x^2+25 x-47)/(a x-2)=-8 x-3-(53)/(a x-2) is...

The equation $\frac{24 x^{2}+25 x-47}{a x-2}=-8 x-3-\frac{53}{a x-2}$ is true for all values of $x \neq \frac{2}{a},$ where $a$ is a constant. What is the value of $a$ ? \begin{equation} \begin{array}{cc}{\text { A) }} & {-16} \\ {\text { B) }} & {-3} \\ {\text { C) }} & {3} \\ {\text { D) }} & {16}\end{array} \end{equation}

The equation $\frac{24 x^{2}+25 x-47}{a x-2}=-8 x-3-\frac{53}{a x-2}$ is
true for all values of $x \neq \frac{2}{a},$ where $a$ is a constant.
What is the value of $a$ ?
\begin{equation}
\begin{array}{cc}{\text { A) }} & {-16} \\ {\text { B) }} & {-3} \\ {\text { C) }} & {3} \\ {\text { D) }} & {16}\end{array}
\end{equation}

Key Concepts

Equating Coefficients

Equating coefficients involves matching the coefficients of the corresponding powers of the variable in two polynomial expressions that are identically equal. After clearing the denominators, the equality must hold for all terms, leading to a system of equations for the coefficients. Solving these equations provides the values of the unknown parameters in the problem.

Rational Expression Identity

A rational expression identity is an equation that is true for all values of the variable (except those that cause division by zero). This implies that after the denominators are removed or cleared, the resulting polynomial expression must hold for every value in the variable's domain. This concept is central when determining parameters that make the equation valid universally.

Clearing Denominators

Clearing denominators is a method used to eliminate fractions in an equation by multiplying every term by the common denominator. This step transforms a rational equation into a polynomial equation, which simplifies the process of finding unknown constants. It is particularly useful when the equation is claimed to be valid for all values in its domain.

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