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$The equation (24x^(2)+25 x-47)/(ax-2)=-8x-3-(53)/(ax-2) is true for all values of , where is a constant. {:[x!=(2)/(a)],[a]:} What is the value of ? a$

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 , where is a constant. $\newline$ $\begin{array}{l} x \neq \frac{2}{a} \\ a \end{array}$ $\newline$ What is the value of ? $\newline$ $a$

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Q. 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 , where is a constant.

\newline

\begin{array}{l} x \neq \frac{2}{a} \\ a \end{array}

\newline

What is the value of ?

\newline

a

Write Equation, Identify Goal: Write down the given equation and identify the goal. $\newline$ The given equation is: $\newline$ $(24x^2 + 25x - 47) / (ax - 2) = -8x - 3 - 53 / (ax - 2)$ $\newline$ We need to find the value of $a$ such that the equation holds true for all values of $x$ , where $x \neq 2/a$ .
Simplify Right-hand Side: Simplify the right-hand side of the equation by combining like terms. $\newline$ $-8x - 3 - \frac{53}{ax - 2}$ can be rewritten as: $\newline$ $-8x - 3 - \left(\frac{53}{ax - 2}\right)$ $\newline$ Since there are no like terms to combine, we move on to the next step.
Eliminate Fraction: Multiply both sides of the equation by $(ax - 2)$ to eliminate the fraction. $(24x^2 + 25x - 47) = (-8x - 3)(ax - 2) - 53$
Expand Equation: Expand the right-hand side of the equation. $\newline$ $(-8x - 3)(ax - 2) - 53 = -8ax^2 + 16x - 3ax + 6 - 53$ $\newline$ Simplify the right-hand side: $\newline$ $-8ax^2 + (16 - 3a)x - 47$
Set Up System: Since the equation must hold for all values of $x$ , the coefficients of the corresponding powers of $x$ must be equal on both sides of the equation. $\newline$ This gives us a system of equations: $\newline$ For the $x^2$ term: $24 = -8a$ $\newline$ For the $x$ term: $25 = 16 - 3a$ $\newline$ For the constant term: $-47 = -47$ (which is already satisfied and does not give us information about $a$ )
Solve for $a$ : Solve the first equation from Step $5$ for $a$ . $24 = -8a$ Divide both sides by $-8$ : $a = -3$
Verify Solution: Verify the solution by substituting $a = -3$ into the second equation from Step $5$ . $\newline$ $25 = 16 - 3a$ $\newline$ Substitute $a = -3$ : $\newline$ $25 = 16 - 3(-3)$ $\newline$ $25 = 16 + 9$ $\newline$ $25 = 25$ $\newline$ This confirms that $a = -3$ is the correct solution.