\begin{aligned} y&=10+16x-x^2\ \ y&=3x+50 \end{aligned} If(x1,y1) and (x2,y2) are distinct solutions to the system of equations shown, what is the sum of the y1 and y2 ?

Question

\begin{aligned}  y&=10+16x-x^2\ \ y&=3x+50 \end{aligned} If(x1,y1)   and (x2,y2)     are distinct solutions to the system of equations shown, what is the sum of the   y1 and  y2  ?

Bytelearn · Accepted Answer

Set Equations Equal: To find the distinct solutions (x1, y1) and (x2, y2) to the system of equations, we need to set the two equations equal to each other and solve for x. \begin{aligned} 10 + 16x - x^2 &= 3x + 50 \ x^2 - 13x + 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: x1 = 5 and x2 = 8.Substitute x1 for y1: We substitute x1 = 5 into one of the original equations to find y1. \begin{aligned} y1 &= 3(5) + 50 \ y1 &= 15 + 50 \ y1 &= 65 \end{aligned}Substitute x2 for y2: We substitute x2 = 8 into one of the original equations to find y2. \begin{aligned} y2 &= 3(8) + 50 \ y2 &= 24 + 50 \ y2 &= 74 \end{aligned}Find Sum of y1 and y2: Now we find the sum of y1 and y2. \begin{aligned} y1 + y2 &= 65 + 74 \ y1 + y2 &= 139 \end{aligned}

\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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\begin{aligned} y&=10+16x-x^2 \\ y&=3x+50 \end{aligned} If (x1,y1)(x_1,y_1)(x1​,y1​) and (x2,y2)(x_2,y_2)(x2​,y2​) are distinct solutions to the system of equations shown, what is the sum of the y1y_1y1​ and y2y_2y2​?

\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 $y_2$ ?