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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 ?

The equation 24x2+25x−47ax−2=−8x−3−53ax−2 \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.\newlinex≠2aa \begin{array}{l} x \neq \frac{2}{a} \\ a \end{array} \newlineWhat is the value of ?

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Q. The equation 24x2+25x−47ax−2=−8x−3−53ax−2 \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.\newlinex≠2aa \begin{array}{l} x \neq \frac{2}{a} \\ a \end{array} \newlineWhat is the value of ?
  1. Write Equation Simplify: Write down the given equation and simplify it.\newlineThe given equation is:\newline(24x2+25x−47)/(ax−2)=−8x−3−53/(ax−2)(24x^2 + 25x - 47)/(ax - 2) = -8x - 3 - 53/(ax - 2)\newlineWe can combine the terms on the right side of the equation:\newline−8x−3−53/(ax−2)=−8x−3−53/(ax−2)-8x - 3 - 53/(ax - 2) = -8x - 3 - 53/(ax - 2)\newlineThis simplifies to:\newline−8x−3−53/(ax−2)-8x - 3 - 53/(ax - 2)
  2. Combine Right Side Terms: Since the equation is true for all values of xx (except x≠2ax \neq \frac{2}{a}), the numerators of the rational expressions on both sides must be equal.\newlineThis means we can set up the equation:\newline24x2+25x−47=(−8x−3)(ax−2)−5324x^2 + 25x - 47 = (-8x - 3)(ax - 2) - 53
  3. Set Up Equation Expand: Expand the right side of the equation.\newline(−8x−3)(ax−2)−53=−8ax2+16x−3ax+6−53(-8x - 3)(ax - 2) - 53 = -8ax^2 + 16x - 3ax + 6 - 53\newlineCombine like terms:\newline−8ax2+(16−3a)x−47-8ax^2 + (16 - 3a)x - 47
  4. Expand Right Side: Since the expressions are equal for all xx, the coefficients of the corresponding powers of xx must be equal.\newlineThis gives us a system of equations:\newlineFor x2x^2 terms: 24=−8a24 = -8a\newlineFor xx terms: 25=16−3a25 = 16 - 3a\newlineFor constant terms: −47=−47-47 = -47 (This is already satisfied and doesn't give us information about aa.)
  5. Solve System Equations: Solve the first equation from the system for aa.24=−8a24 = -8aDivide both sides by −8-8:a=−24−8a = -\frac{24}{-8}a=3a = 3
  6. Check Solution: Check the solution by substituting a=3a = 3 into the second equation.\newline25=16−3a25 = 16 - 3a\newline25=16−3(3)25 = 16 - 3(3)\newline25=16−925 = 16 - 9\newline25=725 = 7\newlineThis is not true, which means there is a math error in the previous steps.

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