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{:[-2-x=(y^(2)-4y+10)/(11)],[-11 x+3y=62]:} If (x_(1),y_(1)) and (x_(2),y_(2)) are two distinct solutions to the system of equations shown, what is the product of the x values of the two solutions x_(1)*x_(2) ? ◻

−2−x=y2−4y+1011-2-x=\frac{y^{2}-4y+10}{11}\newline −11x+3y=62-11 x+3y=62 \newlineIf (x1,y1)(x_{1},y_{1}) and x2,y2)x_{2},y_{2}) are two distinct solutions to the system of equations shown, what is the product of the  x\ x values of the two solutions (x1∗x2)(x_{1}*x_{2})?

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Q. −2−x=y2−4y+1011-2-x=\frac{y^{2}-4y+10}{11}\newline −11x+3y=62-11 x+3y=62 \newlineIf (x1,y1)(x_{1},y_{1}) and x2,y2)x_{2},y_{2}) are two distinct solutions to the system of equations shown, what is the product of the  x\ x values of the two solutions (x1∗x2)(x_{1}*x_{2})?
  1. Solve First Equation: First, let's solve the system of equations using substitution or elimination. I'll start with the first equation:\newline−2−x=y2−4y+1011-2 - x = \frac{y^2 - 4y + 10}{11}\newlineMultiply both sides by 1111 to get rid of the fraction:\newline−22−11x=y2−4y+10-22 - 11x = y^2 - 4y + 10
  2. Rearrange Second Equation: Now let's rearrange the second equation to express yy in terms of xx:
    −11x+3y=62-11x + 3y = 62
    3y=11x+623y = 11x + 62
    y=11x+623y = \frac{11x + 62}{3}
  3. Substitute yy into First: Substitute the expression for yy from the second equation into the first equation:\newline\(-22 - 1111x = \left(\frac{1111x + 6262}{33}\right)^22 - 44\left(\frac{1111x + 6262}{33}\right) + 1010
  4. Expand and Simplify: Now, let's expand and simplify the equation:\newline−22−11x=121x2+1342x+38449−44x+2483+10-22 - 11x = \frac{121x^2 + 1342x + 3844}{9} - \frac{44x + 248}{3} + 10
  5. Combine Like Terms: To combine the terms, we need a common denominator, which is 99: −198−99x=121x2+1342x+3844−132x−744+90-198 - 99x = 121x^2 + 1342x + 3844 - 132x - 744 + 90
  6. Factor or Use Quadratic Formula: Combine like terms:\newline121x2+1342x−132x+3844−744+90=99x+198121x^2 + 1342x - 132x + 3844 - 744 + 90 = 99x + 198\newline121x2+1210x+3190=99x+198121x^2 + 1210x + 3190 = 99x + 198
  7. Calculate Discriminant: Bring all terms to one side to set the equation to zero:\newline121x2+1210x−99x+3190−198=0121x^2 + 1210x - 99x + 3190 - 198 = 0\newline121x2+1111x+2992=0121x^2 + 1111x + 2992 = 0
  8. Plug in Values: Now we need to factor the quadratic equation, but it seems a bit complicated. Let's use the quadratic formula instead:\newlinex=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\newlineHere, a=121a = 121, b=1111b = 1111, and c=2992c = 2992.
  9. Calculate Exact Value: Calculate the discriminant b2−4acb^2 - 4ac:Discriminant=11112−4(121)(2992)\text{Discriminant} = 1111^2 - 4(121)(2992)
  10. Calculate Exact Value: Calculate the discriminant b2−4acb^2 - 4ac:
    Discriminant = 11112−4(121)(2992)1111^2 - 4(121)(2992)Plug in the values and calculate the discriminant:
    Discriminant = 1234321−4(121)(2992)1234321 - 4(121)(2992)
  11. Calculate Exact Value: Calculate the discriminant b2−4acb^2 - 4ac:
    Discriminant = 11112−4(121)(2992)1111^2 - 4(121)(2992)Plug in the values and calculate the discriminant:
    Discriminant = 1234321−4(121)(2992)1234321 - 4(121)(2992)Calculate the exact value of the discriminant:
    Discriminant = 1234321−14484481234321 - 1448448
    Discriminant = (-214127\)

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