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{:[y=-x(x+6)+18],[y=-7x+12]:} If (a,b) is a solution to the system of equations shown and b < 0, what is the value of a ?

y=−x(x+6)+18y=−7x+12 \begin{array}{l} y=-x(x+6)+18 \\ y=-7 x+12 \end{array} \newlineIf (a,b) (a, b) is a solution to the system of equations shown and b<0 b<0 , what is the value of a a ?

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Q. y=−x(x+6)+18y=−7x+12 \begin{array}{l} y=-x(x+6)+18 \\ y=-7 x+12 \end{array} \newlineIf (a,b) (a, b) is a solution to the system of equations shown and b<0 b<0 , what is the value of a a ?
  1. Write equations: Write down the system of equations.\newlineWe have the following system of equations:\newline11) y=−x(x+6)+18y = -x(x + 6) + 18\newline22) y=−7x+12y = -7x + 12
  2. Set equations equal: Set the two equations equal to each other to find the xx-values where their yy-values are the same.\newline−7x+12=−x(x+6)+18-7x + 12 = -x(x + 6) + 18
  3. Expand quadratic equation: Expand the quadratic equation on the right side.−7x+12=−x2−6x+18-7x + 12 = -x^2 - 6x + 18
  4. Rearrange and combine terms: Rearrange the equation to set it to zero and combine like terms. x2+x−6=0x^2 + x - 6 = 0
  5. Factor quadratic equation: Factor the quadratic equation. (x+3)(x−2)=0(x + 3)(x - 2) = 0
  6. Solve for x-values: Solve for the x-values that make the equation true.\newlinex+3=0x + 3 = 0 or x−2=0x - 2 = 0\newlinex=−3x = -3 or x=2x = 2
  7. Determine negative y-value: Determine which x-value corresponds to a negative y-value b<0b < 0.\newlineWe need to plug x=−3x = -3 and x=2x = 2 into either of the original equations to find the corresponding y-values.\newlineLet's use the second equation y=−7x+12y = -7x + 12.\newlineFor x=−3x = -3: y=−7(−3)+12=21+12=33y = -7(-3) + 12 = 21 + 12 = 33\newlineFor x=2x = 2: y=−7(2)+12=−14+12=−2y = -7(2) + 12 = -14 + 12 = -2
  8. Choose xx-value with negative yy: Since we are looking for the xx-value where yy (or bb) is negative, we choose x=2x = 2 because it gives us y=−2y = -2, which is less than 00.\newlineTherefore, the value of aa is 22.

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