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\begin{aligned} y&=10+16x-x^2 \\ y&=3x+50 \end{aligned} If $(x_1,y_1)$ and $(x_2,y_2)$ are distinct solutions to the system of equations shown, what is the sum of the $y_1$ and $y_2$ ?

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Write a linear equation from two points

Full solution

Q. \begin{aligned} y&=10+16x-x^2 \\ y&=3x+50 \end{aligned} If

(x_1,y_1)

and

(x_2,y_2)

are distinct solutions to the system of equations shown, what is the sum of the

y_1

and

y_2

?

Set Equations Equal: To find the distinct solutions $(x_1, y_1)$ and $(x_2, y_2)$ to the system of equations, we need to set the two equations equal to each other and solve for $x$ .\begin{aligned} $10$ + $16$ x - x^ $2$ &= $3$ x + $50$ \x^ $2$ - $13$ x + $40$ &= $0$ \end{aligned}
Factor Quadratic Equation: Now we factor the quadratic equation to find the values of $x$ .\begin{aligned}(x - $5$ )(x - $8$ ) &= $0$ \end{aligned}This gives us two solutions for $x$ : $x_1 = 5$ and $x_2 = 8$ .
Substitute $x_1$ for $y_1$ : We substitute $x_1 = 5$ into one of the original equations to find $y_1$ . $\newline$ \begin{aligned} y_1 &= 3(5) + 50 \ y_1 &= 15 + 50 \ y_1 &= 65 \end{aligned}
Substitute $x^2$ for $y^2$ : We substitute $x^2 = 8$ into one of the original equations to find $y^2$ . $\newline$ \begin{aligned} $\newline$ y^ $2$ &= $3$ ( $8$ ) + $50$ (\newline\)y^ $2$ &= $24$ + $50$ (\newline\)y^ $2$ &= $74$ $\newline$ \end{aligned}
Find Sum of $y_1$ and $y_2$ : Now we find the sum of $y_1$ and $y_2$ . $\newline$ \begin{aligned} y_1 + y_2 &= 65 + 74 \ y_1 + y_2 &= 139 \end{aligned}

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