💡Okay, so I have this problem with complex numbers a, b, c, x, y, z. The equations given are:[a = frac{b + c}{x - 3}, quad b = frac{a + c}{y - 3}, quad c = frac{a + b}{z - 3},]And also, we know that:[xy + xz + yz = 10]and[x + y + z = 6.]We need to find (xyz).Hmm, okay. Let me try to figure this out step by step.First, looking at the equations for a, b, and c, they all relate a, b, c to each other through x, y, z. Maybe I can express x, y, z in terms of a, b, c.Starting with the first equation:[a = frac{b + c}{x - 3}]If I solve for (x - 3), I get:[x - 3 = frac{b + c}{a}]Similarly, for the second equation:[b = frac{a + c}{y - 3}]Solving for (y - 3):[y - 3 = frac{a + c}{b}]And for the third equation:[c = frac{a + b}{z - 3}]Solving for (z - 3):[z - 3 = frac{a + b}{c}]So now I have expressions for (x - 3), (y - 3), and (z - 3) in terms of a, b, c.Let me write them again:[x - 3 = frac{b + c}{a}][y - 3 = frac{a + c}{b}][z - 3 = frac{a + b}{c}]Hmm, maybe I can express (x - 2), (y - 2), (z - 2) instead? Because 2 is a number that comes up in the next step.Wait, why would I think of 2? Maybe because when I see (x - 3), I can write it as (x - 2 - 1), but I'm not sure yet. Let me think.Alternatively, maybe I can add 1 to both sides of each equation. Let's see:From (x - 3 = frac{b + c}{a}), adding 1 to both sides:[x - 2 = frac{b + c}{a} + 1 = frac{b + c + a}{a} = frac{a + b + c}{a}]Ah! That's a good move. Similarly for y and z:From (y - 3 = frac{a + c}{b}), adding 1:[y - 2 = frac{a + c}{b} + 1 = frac{a + c + b}{b} = frac{a + b + c}{b}]And for z:From (z - 3 = frac{a + b}{c}), adding 1:[z - 2 = frac{a + b}{c} + 1 = frac{a + b + c}{c}]Okay, so now we have:[x - 2 = frac{a + b + c}{a}][y - 2 = frac{a + b + c}{b}][z - 2 = frac{a + b + c}{c}]Let me denote (S = a + b + c). Then the above equations become:[x - 2 = frac{S}{a}][y - 2 = frac{S}{b}][z - 2 = frac{S}{c}]So, if I take reciprocals, I get:[frac{1}{x - 2} = frac{a}{S}][frac{1}{y - 2} = frac{b}{S}][frac{1}{z - 2} = frac{c}{S}]Now, if I add these three equations together:[frac{1}{x - 2} + frac{1}{y - 2} + frac{1}{z - 2} = frac{a + b + c}{S} = frac{S}{S} = 1]So, we have:[frac{1}{x - 2} + frac{1}{y - 2} + frac{1}{z - 2} = 1]That's a nice equation. Now, I need to relate this to the given information about x, y, z. We know that (x + y + z = 6) and (xy + yz + zx = 10). We need to find (xyz).Let me denote (p = x - 2), (q = y - 2), (r = z - 2). Then, the equation becomes:[frac{1}{p} + frac{1}{q} + frac{1}{r} = 1]Also, let's express x, y, z in terms of p, q, r:[x = p + 2][y = q + 2][z = r + 2]So, substituting into the given equations:First, (x + y + z = 6):[(p + 2) + (q + 2) + (r + 2) = 6][p + q + r + 6 = 6][p + q + r = 0]Second, (xy + yz + zx = 10):Let's compute (xy + yz + zx):[xy = (p + 2)(q + 2) = pq + 2p + 2q + 4]Similarly,[yz = (q + 2)(r + 2) = qr + 2q + 2r + 4][zx = (r + 2)(p + 2) = rp + 2r + 2p + 4]Adding them together:[xy + yz + zx = (pq + qr + rp) + 2(p + q + r) + 2(q + r + p) + 12]Wait, let me compute each term:First, the products:(xy + yz + zx = (pq + 2p + 2q + 4) + (qr + 2q + 2r + 4) + (rp + 2r + 2p + 4))Combine like terms:- The pq, qr, rp terms: (pq + qr + rp)- The linear terms: 2p + 2q + 2q + 2r + 2r + 2p = 4p + 4q + 4r- The constants: 4 + 4 + 4 = 12So, altogether:[xy + yz + zx = (pq + qr + rp) + 4(p + q + r) + 12]But we know that (p + q + r = 0), so:[xy + yz + zx = (pq + qr + rp) + 0 + 12 = pq + qr + rp + 12]But we are given that (xy + yz + zx = 10), so:[pq + qr + rp + 12 = 10][pq + qr + rp = -2]So now, we have:1. (p + q + r = 0)2. (pq + qr + rp = -2)3. (frac{1}{p} + frac{1}{q} + frac{1}{r} = 1)We need to find (xyz), which is:[xyz = (p + 2)(q + 2)(r + 2)]Let me expand this:[(p + 2)(q + 2)(r + 2) = pqr + 2(pq + pr + qr) + 4(p + q + r) + 8]We already know (pq + pr + qr = -2) and (p + q + r = 0), so substituting:[xyz = pqr + 2(-2) + 4(0) + 8 = pqr - 4 + 0 + 8 = pqr + 4]So, if we can find (pqr), then we can find (xyz).From equation 3:[frac{1}{p} + frac{1}{q} + frac{1}{r} = 1]Let me write this as:[frac{pq + pr + qr}{pqr} = 1]We know (pq + pr + qr = -2), so:[frac{-2}{pqr} = 1][-2 = pqr][pqr = -2]Therefore, substituting back into (xyz):[xyz = (-2) + 4 = 2]Wait, that can't be right because in the initial problem, the given equations lead to (xyz = 14). Did I make a mistake somewhere?Wait, let me check my steps.First, when I defined (p = x - 2), (q = y - 2), (r = z - 2), then (x = p + 2), etc.Then, (x + y + z = 6) becomes (p + q + r + 6 = 6), so (p + q + r = 0). That seems correct.Then, (xy + yz + zx = 10). I expanded it as:[(p + 2)(q + 2) + (q + 2)(r + 2) + (r + 2)(p + 2)]Which is:(pq + 2p + 2q + 4 + qr + 2q + 2r + 4 + rp + 2r + 2p + 4)Combine terms:pq + qr + rp + (2p + 2q + 2q + 2r + 2r + 2p) + (4 + 4 + 4)Which is:pq + qr + rp + 4p + 4q + 4r + 12Since (p + q + r = 0), the linear terms become 0, so:pq + qr + rp + 12 = 10Thus, pq + qr + rp = -2. That seems correct.Then, equation 3:[frac{1}{p} + frac{1}{q} + frac{1}{r} = 1]Which is:[frac{pq + pr + qr}{pqr} = 1]So, (frac{-2}{pqr} = 1), so (pqr = -2). That seems correct.Then, (xyz = (p + 2)(q + 2)(r + 2)). Expanding:(pqr + 2(pq + pr + qr) + 4(p + q + r) + 8)Substituting:(-2 + 2(-2) + 4(0) + 8 = -2 -4 + 0 + 8 = 2)But the initial problem's solution was 14. So, where is the mistake?Wait, maybe I made a mistake in the expansion of (xyz). Let me double-check.[(p + 2)(q + 2)(r + 2)]First, multiply (p + 2)(q + 2):= pq + 2p + 2q + 4Then, multiply by (r + 2):= (pq + 2p + 2q + 4)(r + 2)= pq*r + pq*2 + 2p*r + 2p*2 + 2q*r + 2q*2 + 4*r + 4*2= pqr + 2pq + 2pr + 4p + 2qr + 4q + 4r + 8Now, group like terms:= pqr + 2(pq + pr + qr) + 4(p + q + r) + 8Yes, that's correct. So, substituting:pqr = -22(pq + pr + qr) = 2*(-2) = -44(p + q + r) = 4*0 = 0Plus 8.So total: -2 -4 + 0 + 8 = 2.Hmm, but the initial solution got 14. So, perhaps my substitution is wrong.Wait, let me check the initial substitution.Wait, in the problem statement, the user wrote:"Given the relationships:[x - 3 = frac{b + c}{a}, quad y - 3 = frac{a + c}{b}, quad z - 3 = frac{a + b}{c},]we deduce:[x - 2 = frac{a + b + c}{a}, quad y - 2 = frac{a + b + c}{b}, quad z - 2 = frac{a + b + c}{c}.]"Wait, in my process, I added 1 to both sides of (x - 3 = frac{b + c}{a}), getting (x - 2 = frac{b + c + a}{a} = frac{S}{a}), which is correct.But in the initial solution, they wrote:"Thus, we have:[frac{1}{x - 2} = frac{a}{a + b + c}, quad frac{1}{y - 2} = frac{b}{a + b + c}, quad frac{1}{z - 2} = frac{c}{a + b + c},]and summing these gives:[frac{1}{x - 2} + frac{1}{y - 2} + frac{1}{z - 2} = 1.]"Which is correct.Then, multiplying by ((x - 2)(y - 2)(z - 2)), they get:[(y - 2)(z - 2) + (x - 2)(z - 2) + (x - 2)(y - 2) = (x - 2)(y - 2)(z - 2).]Expanding both sides:Left side: (y - 2)(z - 2) + (x - 2)(z - 2) + (x - 2)(y - 2)= yz - 2y - 2z + 4 + xz - 2x - 2z + 4 + xy - 2x - 2y + 4= xy + yz + xz - 4x - 4y - 4z + 12Right side: (x - 2)(y - 2)(z - 2)= xyz - 2xy - 2xz - 2yz + 4x + 4y + 4z - 8So, setting left = right:xy + yz + xz - 4x - 4y - 4z + 12 = xyz - 2xy - 2xz - 2yz + 4x + 4y + 4z - 8Bring all terms to left side:xy + yz + xz - 4x - 4y - 4z + 12 - xyz + 2xy + 2xz + 2yz - 4x - 4y - 4z + 8 = 0Combine like terms:(1 + 2)xy + (1 + 2)yz + (1 + 2)xz - (4 + 4)x - (4 + 4)y - (4 + 4)z + (12 + 8) - xyz = 0So:3xy + 3yz + 3xz - 8x - 8y - 8z + 20 - xyz = 0Factor:3(xy + yz + xz) - 8(x + y + z) + 20 - xyz = 0We know that (xy + yz + xz = 10) and (x + y + z = 6), so substitute:3*10 - 8*6 + 20 - xyz = 0Compute:30 - 48 + 20 - xyz = 0(30 + 20) - 48 - xyz = 050 - 48 - xyz = 02 - xyz = 0Thus, xyz = 2.Wait, but the initial solution said xyz = 14. So, which one is correct?Wait, in the initial problem, the user wrote:"Thus, we have:[frac{1}{x - 2} = frac{a}{a + b + c}, quad frac{1}{y - 2} = frac{b}{a + b + c}, quad frac{1}{z - 2} = frac{c}{a + b + c},]and summing these gives:[frac{1}{x - 2} + frac{1}{y - 2} + frac{1}{z - 2} = 1.]Multiplying by ((x - 2)(y - 2)(z - 2)) and expanding, we get:[(y - 2)(z - 2) + (x - 2)(z - 2) + (x - 2)(y - 2) = (x - 2)(y - 2)(z - 2).]Expanding and rearranging, we obtain:[xy + xz + yz - 2(x + y + z) + 4 = xyz - (xy + xz + yz) + (x + y + z) - 2,]so:[xyz = 2(xy + xz + yz) - 3(x + y + z) + 6 = 2 times 10 - 3 times 6 + 6 = boxed{14}.]"Wait, in their expansion, they have:Left side: (y - 2)(z - 2) + (x - 2)(z - 2) + (x - 2)(y - 2) = (x - 2)(y - 2)(z - 2)They expanded the left side as:xy + xz + yz - 2(x + y + z) + 4But when I expanded it earlier, I got:xy + yz + xz - 4x - 4y - 4z + 12Wait, so which is correct?Wait, let me compute (y - 2)(z - 2):= yz - 2y - 2z + 4Similarly, (x - 2)(z - 2) = xz - 2x - 2z + 4And (x - 2)(y - 2) = xy - 2x - 2y + 4Adding all three:(yz - 2y - 2z + 4) + (xz - 2x - 2z + 4) + (xy - 2x - 2y + 4)= xy + yz + xz - 2x - 2y - 2z - 2x - 2y - 2z + 4 + 4 + 4Wait, no, that's not correct.Wait, let's do it term by term:First term: yz - 2y - 2z + 4Second term: xz - 2x - 2z + 4Third term: xy - 2x - 2y + 4Now, adding all together:- yz + xz + xyThen, the linear terms:-2y -2z -2x -2z -2x -2yWhich is:-2x -2x -2y -2y -2z -2z = -4x -4y -4zConstants:4 + 4 + 4 = 12So, total:xy + yz + xz -4x -4y -4z + 12So, the left side is:xy + yz + xz -4x -4y -4z + 12The right side is:(x - 2)(y - 2)(z - 2) = xyz - 2xy - 2xz - 2yz + 4x + 4y + 4z - 8So, setting left = right:xy + yz + xz -4x -4y -4z + 12 = xyz - 2xy - 2xz - 2yz + 4x + 4y + 4z - 8Bring all terms to left:xy + yz + xz -4x -4y -4z + 12 - xyz + 2xy + 2xz + 2yz -4x -4y -4z +8 = 0Wait, no, actually, subtract right side from both sides:Left - Right = 0So:(xy + yz + xz -4x -4y -4z + 12) - (xyz - 2xy - 2xz - 2yz + 4x + 4y + 4z - 8) = 0Compute term by term:xy - (-2xy) = 3xyyz - (-2yz) = 3yzxz - (-2xz) = 3xz-4x - 4x = -8x-4y -4y = -8y-4z -4z = -8z12 - (-8) = 20- xyzSo, altogether:3xy + 3yz + 3xz -8x -8y -8z +20 -xyz =0Which is:3(xy + yz + xz) -8(x + y + z) +20 -xyz=0Given that (xy + yz + xz =10) and (x + y + z=6), substitute:3*10 -8*6 +20 -xyz=030 -48 +20 -xyz=0(30 +20) -48 -xyz=050 -48 -xyz=02 -xyz=0Thus, xyz=2So, according to my calculation, xyz=2But in the initial problem, the solution got 14. So, which one is correct?Wait, perhaps the initial solution made a mistake in the expansion.Looking back at the initial solution:"Expanding and rearranging, we obtain:[xy + xz + yz - 2(x + y + z) + 4 = xyz - (xy + xz + yz) + (x + y + z) - 2,]so:[xyz = 2(xy + xz + yz) - 3(x + y + z) + 6 = 2 times 10 - 3 times 6 + 6 = boxed{14}.]"Wait, in their expansion, they have:Left side: (y - 2)(z - 2) + (x - 2)(z - 2) + (x - 2)(y - 2) = (x - 2)(y - 2)(z - 2)They expanded the left side as:xy + xz + yz - 2(x + y + z) + 4But according to my calculation, the left side is:xy + yz + xz -4x -4y -4z + 12So, their expansion is incorrect. They only subtracted 2(x + y + z) and added 4, but in reality, it's -4(x + y + z) +12.Therefore, their mistake led them to an incorrect result of 14, whereas the correct answer is 2.But wait, in my substitution earlier, when I set p = x -2, etc., I got xyz=2. But in the initial problem, the user wrote that the answer is 14.Wait, perhaps I made a mistake in substitution.Wait, let me check again.I set p = x -2, q = y -2, r = z -2.Then, x = p +2, y = q +2, z = r +2.Then, x + y + z =6 becomes p + q + r +6=6, so p + q + r=0.Then, xy + yz + zx=10.I expanded it as:(p +2)(q +2) + (q +2)(r +2) + (r +2)(p +2) =10Which is:pq +2p +2q +4 + qr +2q +2r +4 + rp +2r +2p +4=10Combine terms:pq + qr + rp +4p +4q +4r +12=10Since p + q + r=0, 4(p + q + r)=0, so:pq + qr + rp +12=10Thus, pq + qr + rp= -2Then, from the reciprocal equation:1/p +1/q +1/r=1Which is (pq + pr + qr)/(pqr)=1So, (-2)/pqr=1 => pqr=-2Then, xyz=(p +2)(q +2)(r +2)=pqr +2(pq + pr + qr) +4(p + q + r)+8= -2 +2*(-2)+0 +8= -2 -4 +8=2So, according to this substitution, xyz=2But in the initial problem, the user's solution got 14, which seems incorrect.Wait, but perhaps the initial problem's solution is wrong because of incorrect expansion.Alternatively, perhaps I made a mistake in substitution.Wait, let me think differently.Let me consider the initial equations:a = (b + c)/(x -3)Similarly for b and c.Let me assume that a = b = c. Maybe that's a possible case.If a = b = c, then:a = (a + a)/(x -3) => a = 2a/(x -3)Assuming a ≠0, we can divide both sides by a:1 = 2/(x -3) => x -3=2 =>x=5Similarly, y=5 and z=5But then, x + y + z=15, which contradicts x + y + z=6.So, a, b, c cannot all be equal.Alternatively, maybe two of them are equal.But perhaps that's complicating.Alternatively, let me consider the equations:From a = (b + c)/(x -3)Similarly,b = (a + c)/(y -3)c = (a + b)/(z -3)Let me denote:Let me write these as:a(x -3) = b + cb(y -3) = a + cc(z -3) = a + bSo, we have:1. a(x -3) = b + c2. b(y -3) = a + c3. c(z -3) = a + bLet me add all three equations:a(x -3) + b(y -3) + c(z -3) = 2(a + b + c)Let me denote S = a + b + cThen,a(x -3) + b(y -3) + c(z -3) = 2SLet me express a, b, c from the original equations.From equation 1: a = (b + c)/(x -3) = (S - a)/(x -3)So,a(x -3) = S - aThus,a(x -3 +1)=Sa(x -2)=SSimilarly,From equation 2: b(y -3)=a + c = S - bThus,b(y -3 +1)=Sb(y -2)=SSimilarly,From equation3: c(z -3)=a + b = S - cThus,c(z -3 +1)=Sc(z -2)=SSo, we have:a = S/(x -2)b = S/(y -2)c = S/(z -2)Thus, S = a + b + c = S/(x -2) + S/(y -2) + S/(z -2)Divide both sides by S (since S ≠0, as a,b,c are non-zero):1 = 1/(x -2) + 1/(y -2) + 1/(z -2)Which is the same equation as before.Thus, 1/(x -2) +1/(y -2)+1/(z -2)=1Which is consistent.So, as before, we can set p =x -2, q = y -2, r = z -2Thus, 1/p +1/q +1/r=1And, x + y + z=6 => p + q + r=0And, xy + yz + zx=10We need to find xyz.Earlier, I found that xyz=2, but the initial solution said 14.Wait, perhaps the initial solution made a mistake in the expansion.Alternatively, perhaps I made a mistake in the substitution.Wait, let me compute xyz directly.From p + q + r=0pq + pr + qr=-2pqr=-2Then, xyz=(p +2)(q +2)(r +2)=pqr +2(pq + pr + qr)+4(p + q + r)+8= (-2) +2*(-2)+4*0 +8= -2 -4 +0 +8=2Thus, xyz=2Therefore, the correct answer is 2.But the initial solution said 14, which is incorrect.So, perhaps the initial solution made a mistake in the expansion.Therefore, the correct answer is 2.But wait, in the initial problem, the user wrote:"Expanding and rearranging, we obtain:[xy + xz + yz - 2(x + y + z) + 4 = xyz - (xy + xz + yz) + (x + y + z) - 2,]so:[xyz = 2(xy + xz + yz) - 3(x + y + z) + 6 = 2 times 10 - 3 times 6 + 6 = boxed{14}.]"But in reality, the correct expansion gives:3(xy + yz + xz) -8(x + y + z) +20 -xyz=0Thus, 3*10 -8*6 +20 -xyz=0 =>30 -48 +20 -xyz=0 =>2 -xyz=0 =>xyz=2Therefore, the correct answer is 2.So, the initial solution was wrong because of incorrect expansion.Therefore, the correct answer is 2.But wait, in my substitution, I got xyz=2, but in the initial problem, the user wrote that the answer is 14.So, perhaps the initial problem's solution was wrong.Therefore, the correct answer is 2.But to be sure, let me check with specific numbers.Suppose p, q, r are roots of the equation t^3 - (p + q + r)t^2 + (pq + pr + qr)t - pqr=0Given p + q + r=0, pq + pr + qr=-2, pqr=-2Thus, the equation is t^3 + (-2)t - (-2)= t^3 -2t +2=0Let me find roots of t^3 -2t +2=0Try t=-1: (-1)^3 -2*(-1)+2= -1 +2 +2=3≠0t=1:1 -2 +2=1≠0t=2:8 -4 +2=6≠0t= -2: -8 -(-4)+2= -8 +4 +2=-2≠0Thus, no rational roots. So, p, q, r are roots of t^3 -2t +2=0But regardless, the product pqr=-2, so xyz=2.Thus, the correct answer is 2.Therefore, the initial solution's answer of 14 is incorrect.

💡Okay, so I have this problem here where Sarah's bowling score was 60 points more than Greg's, and together their scores add up to 260. I need to find out what Sarah's score was. Hmm, let me think about how to approach this.First, I should probably define some variables to represent their scores. Let's say Sarah's score is S and Greg's score is G. That seems straightforward. Now, according to the problem, Sarah's score was 60 points more than Greg's. So, I can write that as an equation: S = G + 60. That makes sense because if Greg scored G points, Sarah scored 60 more than that.Next, the problem also says that the sum of their two scores was 260. So, that means if I add Sarah's score and Greg's score together, it equals 260. In equation form, that would be S + G = 260. Okay, so now I have two equations:1. S = G + 602. S + G = 260Now, I need to solve these equations to find out what S is. Since I already have S expressed in terms of G in the first equation, I can substitute that into the second equation. That way, I can solve for G first and then find S.So, substituting S from the first equation into the second equation, I get:(G + 60) + G = 260Let me simplify that. Combining like terms, G + G is 2G, and then we have +60. So, the equation becomes:2G + 60 = 260Now, I need to solve for G. To do that, I'll subtract 60 from both sides of the equation to isolate the term with G:2G + 60 - 60 = 260 - 60Simplifying that, I get:2G = 200Now, to find the value of G, I'll divide both sides by 2:2G / 2 = 200 / 2Which simplifies to:G = 100So, Greg's score is 100. Now, I need to find Sarah's score, which is 60 points more than Greg's. Using the first equation, S = G + 60, and substituting G = 100:S = 100 + 60Which means:S = 160Therefore, Sarah's score is 160. Let me just double-check to make sure this makes sense. If Sarah scored 160 and Greg scored 100, then together they scored 160 + 100 = 260, which matches the total given in the problem. Also, 160 is indeed 60 points more than 100, so everything checks out.I think that's it. I've defined the variables, set up the equations based on the problem statement, solved for Greg's score, and then used that to find Sarah's score. The numbers add up correctly, so I'm confident that Sarah's score is 160.

💡First, I need to determine the number of students who prefer tennis at both North High School and South High School.At North High School:- Total number of students = 1800- Percentage of students preferring tennis = 25%- Number of students preferring tennis = 1800 * 0.25 = 450 studentsAt South High School:- Total number of students = 2700- Percentage of students preferring tennis = 35%- Number of students preferring tennis = 2700 * 0.35 = 945 studentsNext, I'll calculate the total number of students who prefer tennis from both schools:- Total students preferring tennis = 450 + 945 = 1395 studentsThen, I'll find the combined total number of students at both schools:- Combined total number of students = 1800 + 2700 = 4500 studentsFinally, I'll determine the combined percentage of students preferring tennis:- Percentage = (1395 / 4500) * 100% = 31%Therefore, the combined percentage of students preferring tennis from both schools is 31%.

💡First, identify the number of marbles Connie has left after giving some to Juan, which is 70 marbles.Next, determine how many marbles Connie gave to Juan, which is 73 marbles.To find out how many marbles Connie had initially, add the marbles she gave away to the marbles she has left.70 marbles (left) + 73 marbles (given) = 143 marblesTherefore, Connie initially had 143 marbles before giving any to Juan.

💡First, I need to determine the probability of drawing a blue gumball, denoted as ( P(B) ). Jean drew two blue gumballs in a row with replacement, and the probability of this event is given as ( frac{25}{49} ).Since the draws are independent and with replacement, the probability of drawing a blue gumball on each draw is the same. Therefore, the probability of drawing two blue gumballs in a row is:[P(B) times P(B) = left( P(B) right)^2 = frac{25}{49}]To find ( P(B) ), I take the square root of both sides:[P(B) = sqrt{frac{25}{49}} = frac{5}{7}]Next, since there are only blue and pink gumballs in the jar, the probabilities of drawing a blue or a pink gumball must add up to 1:[P(B) + P(P) = 1]Substituting the value of ( P(B) ) that I found:[frac{5}{7} + P(P) = 1]Solving for ( P(P) ):[P(P) = 1 - frac{5}{7} = frac{7}{7} - frac{5}{7} = frac{2}{7}]Therefore, the probability that the next gumball Jean draws will be pink is ( frac{2}{7} ).

💡Okay, so I have this problem here: "There is exactly one isosceles triangle that has a side of length 10 and a side of length 22. What is the perimeter of this triangle?" Hmm, let me try to figure this out step by step.First, I know that an isosceles triangle has at least two sides that are equal in length. So, if the triangle has sides of length 10 and 22, there are two possibilities for the third side: either it's another 10 or another 22. That gives us two potential triangles: one with sides 10, 10, and 22, and another with sides 10, 22, and 22.But wait, I remember something about the triangle inequality theorem. It says that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. So, I need to check if both of these potential triangles satisfy that condition.Let's start with the first possibility: sides of 10, 10, and 22. If I add the two shorter sides, 10 + 10, that equals 20. Now, 20 is less than 22, which means this combination doesn't satisfy the triangle inequality theorem. So, a triangle with sides 10, 10, and 22 isn't possible because the two shorter sides aren't long enough to form a triangle with the longest side.Okay, so that leaves us with the second possibility: sides of 10, 22, and 22. Let's check the triangle inequality for this set. First, 10 + 22 equals 32, which is greater than 22. Next, 22 + 22 equals 44, which is greater than 10. Finally, 10 + 22 equals 32, which is greater than 22. All three combinations satisfy the triangle inequality theorem, so this triangle is possible.Now that I've determined the sides of the triangle are 10, 22, and 22, I can find the perimeter by adding all the sides together. So, 10 + 22 + 22 equals 54. Therefore, the perimeter of the triangle is 54.Wait a second, let me just double-check to make sure I didn't make any mistakes. I considered both possibilities for the equal sides, applied the triangle inequality theorem correctly, and the only valid triangle is with sides 10, 22, and 22. Adding those up gives 54, which seems right. I don't think I missed anything here.