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Which of the following represents all solutions (x,y) to the system of equations shown? y+x=6 y=x^(2)-2x-6 Choose 1 answer: (A) (-4,10) and (3,3) (B) (4,2) and (-3,9) (c) (4,-3) (D) (-4,3)

Which of the following represents all solutions (x,y) (x, y) to the system of equations shown?\newliney+x=6 y+x=6 \newliney=x2−2x−6 y=x^{2}-2 x-6 \newlineChoose 11 answer:\newline(A) (−4,10) (-4,10) and (3,3) (3,3) \newline(B) (4,2) (4,2) and (−3,9) (-3,9) \newline(C) (4,−3) (4,-3) \newline(D) (−4,3) (-4,3)

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Full solution

Q. Which of the following represents all solutions (x,y) (x, y) to the system of equations shown?\newliney+x=6 y+x=6 \newliney=x2−2x−6 y=x^{2}-2 x-6 \newlineChoose 11 answer:\newline(A) (−4,10) (-4,10) and (3,3) (3,3) \newline(B) (4,2) (4,2) and (−3,9) (-3,9) \newline(C) (4,−3) (4,-3) \newline(D) (−4,3) (-4,3)
  1. Write equations: Write down the system of equations to be solved.\newlineThe system of equations is:\newliney+x=6y + x = 6\newliney=x2−2x−6y = x^2 - 2x - 6
  2. Set equations equal: Since both equations equal yy, set them equal to each other to find the xx-values that satisfy both equations.x2−2x−6=x+6x^2 - 2x - 6 = x + 6
  3. Rearrange and solve: Rearrange the equation to set it to zero and solve for xx.x2−2x−6−x−6=0x^2 - 2x - 6 - x - 6 = 0x2−3x−12=0x^2 - 3x - 12 = 0
  4. Factor quadratic equation: Factor the quadratic equation to find the values of xx.(x−4)(x+3)=0(x - 4)(x + 3) = 0
  5. Solve for x: Solve for x by setting each factor equal to zero.\newlinex−4=0x - 4 = 0 or x+3=0x + 3 = 0\newlinex=4x = 4 or x=−3x = -3
  6. Find y-values: Plug the x-values back into the original equations to find the corresponding y-values.\newlineFor x=4x = 4:\newliney+4=6y + 4 = 6\newliney=6−4y = 6 - 4\newliney=2y = 2\newlineFor x=−3x = -3:\newliney−3=6y - 3 = 6\newliney=6+3y = 6 + 3\newliney=9y = 9
  7. Write solution pairs: Write down the solution pairs (x,y)(x,y) for the system of equations.\newlineThe solution pairs are (4,2)(4,2) and (−3,9)(-3,9).

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