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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−xamp;=y2−4y+1011−11x+3yamp;=62 \begin{aligned} -2-x & =\frac{y^{2}-4 y+10}{11} \\ -11 x+3 y & =62 \end{aligned} \newlineIf (x1,y1) \left(x_{1}, y_{1}\right) and (x2,y2) \left(x_{2}, y_{2}\right) 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} \cdot x_{2} ?

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Q. −2−x=y2−4y+1011−11x+3y=62 \begin{aligned} -2-x & =\frac{y^{2}-4 y+10}{11} \\ -11 x+3 y & =62 \end{aligned} \newlineIf (x1,y1) \left(x_{1}, y_{1}\right) and (x2,y2) \left(x_{2}, y_{2}\right) 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} \cdot x_{2} ?
  1. Write Equations: Write down the given system of equations.\newlineThe system of equations is:\newline−2−x=y2−4y+1011-2 - x = \frac{y^2 - 4y + 10}{11}\newline−11x+3y=62-11x + 3y = 62
  2. Eliminate Fraction: Multiply the first equation by 1111 to eliminate the fraction.11(−2−x)=y2−4y+1011(-2 - x) = y^2 - 4y + 10−22−11x=y2−4y+10-22 - 11x = y^2 - 4y + 10
  3. Express yy in xx: Rearrange the second equation to express yy in terms of xx.
    -11x+3y=6211x + 3y = 62
    3y=11x+623y = 11x + 62
    y=11x+623y = \frac{11x + 62}{3}
  4. Substitute in Equation: Substitute the expression for yy from Step 33 into the equation from Step 22.\newline−22−11x=(11x+623)2−4(11x+623)+10-22 - 11x = \left(\frac{11x + 62}{3}\right)^2 - 4\left(\frac{11x + 62}{3}\right) + 10
  5. Expand and Distribute: Expand the square and distribute the 44 in the equation from Step 44.−22−11x=121x2+1342x+38449−44x+2483+10-22 - 11x = \frac{121x^2 + 1342x + 3844}{9} - \frac{44x + 248}{3} + 10
  6. Eliminate Fractions: Multiply through by 99 to eliminate the fractions.−198−99x=121x2+1342x+3844−3(44x+248)+90-198 - 99x = 121x^2 + 1342x + 3844 - 3(44x + 248) + 90
  7. Combine Like Terms: Distribute the 33 and combine like terms.−198−99x=121x2+1342x+3844−132x−744+90-198 - 99x = 121x^2 + 1342x + 3844 - 132x - 744 + 90
  8. Simplify Equation: Simplify the equation.\newline−198−99x=121x2+1210x+3190-198 - 99x = 121x^2 + 1210x + 3190
  9. Form Quadratic Equation: Move all terms to one side to form a quadratic equation. 121x2+1309x+3388=0121x^2 + 1309x + 3388 = 0
  10. Calculate Discriminant: Factor the quadratic equation, if possible, to find the xx values.\newlineThis quadratic does not factor nicely, so we will use the quadratic formula to find the xx values.\newlinex=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\newlineHere, a=121a = 121, b=1309b = 1309, and c=3388c = 3388.
  11. Calculate Discriminant: Calculate the discriminant b2−4acb^2 - 4ac to ensure that there are two distinct real solutions.\newlineDiscriminant = 13092−4(121)(3388)1309^2 - 4(121)(3388)
  12. Find Product of Roots: Calculate the discriminant.\newlineDiscriminant = 1713481−16366081713481 - 1636608\newlineDiscriminant = 7687376873\newlineSince the discriminant is positive, there are two distinct real solutions.
  13. Simplify Product: Use Vieta's formulas to find the product of the roots without actually solving for the roots.\newlineThe product of the roots of a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 is given by ca\frac{c}{a}.\newlineSo, x1×x2=ca=3388121x_1 \times x_2 = \frac{c}{a} = \frac{3388}{121}
  14. Simplify Product: Use Vieta's formulas to find the product of the roots without actually solving for the roots.\newlineThe product of the roots of a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 is given by ca\frac{c}{a}.\newlineSo, x1×x2=ca=3388121x_1 \times x_2 = \frac{c}{a} = \frac{3388}{121} Simplify the product of the roots.\newlinex1×x2=3388121x_1 \times x_2 = \frac{3388}{121}\newline$x_1 \times x_2 = \(28\)

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