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y=-3x (y-7)/(6)=(3x^(2)+x-5)/(2) If (x_(1),y_(1)) and (x_(2),y_(2)) are the distinct solutions to the system of equations shown, what is the sum of the y_(1)+y_(2) ?

y=−3x y=-3 x \newliney−76=3x2+x−52 \frac{y-7}{6}=\frac{3 x^{2}+x-5}{2} \newlineIf (x1,y1) \left(x_{1}, y_{1}\right) and (x2,y2) \left(x_{2}, y_{2}\right) are the distinct solutions to the system of equations shown, what is the sum of the y1+y2 y_{1}+y_{2} ?

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Q. y=−3x y=-3 x \newliney−76=3x2+x−52 \frac{y-7}{6}=\frac{3 x^{2}+x-5}{2} \newlineIf (x1,y1) \left(x_{1}, y_{1}\right) and (x2,y2) \left(x_{2}, y_{2}\right) are the distinct solutions to the system of equations shown, what is the sum of the y1+y2 y_{1}+y_{2} ?
  1. Write Equations: Write down the given system of equations.\newlineWe have the following system:\newliney=−3xy = -3x\newliney−76=3x2+x−52\frac{y - 7}{6} = \frac{3x^2 + x - 5}{2}
  2. Substitute yy in Second Equation: Substitute the expression for yy from the first equation into the second equation.\newlineSince y=−3xy = -3x, we can replace yy in the second equation with −3x-3x:\newline(−3x−7)6=(3x2+x−5)2\frac{(-3x - 7)}{6} = \frac{(3x^2 + x - 5)}{2}
  3. Eliminate Fraction: Multiply both sides of the equation by 66 to eliminate the fraction on the left side.\newline6×(−3x−76)=6×(3x2+x−52)6 \times \left(\frac{-3x - 7}{6}\right) = 6 \times \left(\frac{3x^2 + x - 5}{2}\right)\newlineThis simplifies to:\newline−3x−7=3×(3x2+x−5)-3x - 7 = 3 \times (3x^2 + x - 5)
  4. Multiply Out: Multiply out the right side of the equation. −3x−7=9x2+3x−15-3x - 7 = 9x^2 + 3x - 15
  5. Rearrange to Quadratic Equation: Rearrange the equation to set it to zero and form a quadratic equation.\newline9x2+3x−15+3x+7=09x^2 + 3x - 15 + 3x + 7 = 0\newline9x2+6x−8=09x^2 + 6x - 8 = 0
  6. Factor or Use Quadratic Formula: Factor the quadratic equation, if possible, or use the quadratic formula to find the roots.\newlineThe quadratic equation 9x2+6x−89x^2 + 6x - 8 does not factor easily, so we will use the quadratic formula:\newlinex=−b±b2−4ac2ax = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\newlineHere, a=9a = 9, b=6b = 6, and c=−8c = -8.
  7. Calculate Discriminant: Calculate the discriminant b2−4acb^2 - 4ac to ensure that there are real solutions.\newlineDiscriminant = b2−4acb^2 - 4ac = 62−4(9)(−8)6^2 - 4(9)(-8) = 36+28836 + 288 = 324{324}\newlineSince the discriminant is positive, there are two distinct real solutions.
  8. Calculate Roots: Calculate the roots using the quadratic formula.\newlinex1,2=−6±3242⋅9x_{1,2} = \frac{{-6 \pm \sqrt{324}}}{{2 \cdot 9}}\newlinex1,2=−6±1818x_{1,2} = \frac{{-6 \pm 18}}{{18}}\newlinex1=1218=23x_{1} = \frac{{12}}{{18}} = \frac{2}{3}\newlinex2=−2418=−43x_{2} = \frac{{-24}}{{18}} = -\frac{4}{3}
  9. Find Corresponding y-values: Find the corresponding y-values using the first equation y=−3xy = -3x.y(1)=−3×(23)=−2y_{(1)} = -3 \times \left(\frac{2}{3}\right) = -2y(2)=−3×(−43)=4y_{(2)} = -3 \times \left(-\frac{4}{3}\right) = 4
  10. Calculate Sum of y-values: Calculate the sum of y1+y2y_{1} + y_{2}.\newliney1+y2=−2+4=2y_{1} + y_{2} = -2 + 4 = 2

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