7y^(2)=50 x-150 y=(3-x)/(2) If (x_(1),y_(1)) and (x_(2),y_(2)) are distinct solutions to the system of equations shown, what is the product of the y_(1) and y_(2) ?

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7y^(2)=50 x-150  y=(3-x)/(2) If  (x_(1),y_(1)) and  (x_(2),y_(2)) are distinct solutions to the system of equations shown, what is the product of the  y_(1) and  y_(2) ?

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Equations: We have two equations: 1) 7y^(2) = 50x - 150 2) y = (3 - x)/2  To find the solutions to the system, we can substitute the second equation into the first one to eliminate y and solve for x.Substitution: Substitute y from the second equation into the first equation: 7((3 - x)/2)^(2) = 50x - 150Expand and Simplify: Expand and simplify the equation: 7(9 - 6x + x^2)/4 = 50x - 150 Multiply both sides by 4 to get rid of the denominator: 7(9 - 6x + x^2) = 200x - 600Rearrange the Equation: Distribute the 7 on the left side of the equation: 63 - 42x + 7x^2 = 200x - 600Quadratic Formula: Rearrange the equation to form a quadratic equation: 7x^2 - 42x - 200x + 63 + 600 = 0 7x^2 - 242x + 663 = 0Calculate Discriminant: We need to solve the quadratic equation for x. However, the quadratic does not factor easily, so we will use the quadratic formula: x = [-b ± sqrt(b^2 - 4ac)] / (2a) where a = 7, b = -242, and c = 663.Calculate Product of Roots: Calculate the discriminant (b^2 - 4ac): Discriminant = (-242)^2 - 4(7)(663) Discriminant = 58564 - 18624 Discriminant = 39940Calculate Product of y: Since the discriminant is positive, we have two distinct real solutions for x. Now we calculate the solutions using the quadratic formula: x_(1,2) = [242 ± sqrt(39940)] / 14Simplify Expression: We are interested in the product of y_(1) and y_(2), not the actual values of x_(1) and x_(2). According to Vieta's formulas, for a quadratic equation ax^2 + bx + c = 0, the product of the roots x_(1) * x_(2) is c/a. In our case, c = 663 and a = 7.Final Calculation: Calculate the product of the roots x_(1) * x_(2): x_(1) * x_(2) = c/a = 663/7Now we need to find the product of y_(1) and y_(2). Since y = (3 - x)/2, the product y_(1) * y_(2) will be [(3 - x_(1))/2] * [(3 - x_(2))/2]. According to Vieta's formulas, x_(1) + x_(2) = -b/a. We can use this to find 3 - (x_(1) + x_(2)).Calculate 3 - (x_(1) + x_(2)) using Vieta's formulas: 3 - (x_(1) + x_(2)) = 3 - (-b/a) = 3 - (-242/7) = 3 + 242/7Simplify the expression: 3 + 242/7 = (21 + 242)/7 = 263/7Now we can find the product of y_(1) and y_(2) using the expression we found for 3 - (x_(1) + x_(2)): y_(1) * y_(2) = [(263/7)/2] * [(263/7)/2]Simplify the product of y_(1) and y_(2): y_(1) * y_(2) = (263/14) * (263/14) = (263^2)/(14^2)Calculate the final value: y_(1) * y_(2) = (263^2)/(196) y_(1) * y_(2) = 69169/196

$7 y^{2}=50 x-150$ $\newline$ $y=\frac{3-x}{2}$ $\newline$ If $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ are distinct solutions to the system of equations shown, what is the product of the $y_{1}$ and $y_{2}$ ?

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