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

$\begin{cases} -2-x=\frac{y^{2}-4y+10}{11}, \ -11 x+3y=62 \end{cases}$ 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 $\newline x$ values of the two solutions $x_{1}*x_{2})$ ?

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Algebra 1

Compare linear, exponential, and quadratic growth

Full solution

\begin{cases} -2-x=\frac{y^{2}-4y+10}{11}, \ -11 x+3y=62 \end{cases}

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 $\newline x$ values of the two solutions

x_{1}*x_{2})

Solve First Equation: First, let's solve the system of equations using substitution or elimination. I'll start with the first equation: $\newline$ $-2 - x = \frac{y^2 - 4y + 10}{11}$ $\newline$ Multiply both sides by $11$ to get rid of the fraction: $\newline$ $-22 - 11x = y^2 - 4y + 10$
Rearrange Second Equation: Now let's rearrange the second equation to express $y$ in terms of $x$ :
$-11x + 3y = 62$
$3y = 11x + 62$
$y = \frac{11x + 62}{3}$
Substitute $y$ into First: Substitute the expression for $y$ from the second equation into the first equation: $\newline$ \(-22 - $11$ x = \left(\frac{ $11$ x + $62$ }{ $3$ }\right)^ $2$ - $4$ \left(\frac{ $11$ x + $62$ }{ $3$ }\right) + $10$
Expand and Simplify: Now, let's expand and simplify the equation: $\newline$ $-22 - 11x = \frac{121x^2 + 1342x + 3844}{9} - \frac{44x + 248}{3} + 10$
Combine Like Terms: To combine the terms, we need a common denominator, which is $9$ : $-198 - 99x = 121x^2 + 1342x + 3844 - 132x - 744 + 90$
Factor or Use Quadratic Formula: Combine like terms: $\newline$ $121x^2 + 1342x - 132x + 3844 - 744 + 90 = 99x + 198$ $\newline$ $121x^2 + 1210x + 3190 = 99x + 198$
Calculate Discriminant: Bring all terms to one side to set the equation to zero: $\newline$ $121x^2 + 1210x - 99x + 3190 - 198 = 0$ $\newline$ $121x^2 + 1111x + 2992 = 0$
Plug in Values: Now we need to factor the quadratic equation, but it seems a bit complicated. Let's use the quadratic formula instead: $\newline$ $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $\newline$ Here, $a = 121$ , $b = 1111$ , and $c = 2992$ .
Calculate Exact Value: Calculate the discriminant $b^2 - 4ac$ : $\text{Discriminant} = 1111^2 - 4(121)(2992)$
Calculate Exact Value: Calculate the discriminant $b^2 - 4ac$ :
Discriminant = $1111^2 - 4(121)(2992)$ Plug in the values and calculate the discriminant:
Discriminant = $1234321 - 4(121)(2992)$
Calculate Exact Value: Calculate the discriminant $b^2 - 4ac$ :
Discriminant = $1111^2 - 4(121)(2992)$ Plug in the values and calculate the discriminant:
Discriminant = $1234321 - 4(121)(2992)$ Calculate the exact value of the discriminant:
Discriminant = $1234321 - 1448448$
Discriminant = (-214127\)