💡Okay, so I have this problem here: I need to find the ratio of a to b given that (6a - 5b)/(8a - 3b) equals 2/7. Hmm, ratios can sometimes be tricky, but let me see if I can work through this step by step.First, I remember that when you have a proportion like this, you can cross-multiply to get rid of the fractions. So, if I have (6a - 5b)/(8a - 3b) = 2/7, I can cross-multiply and get 7*(6a - 5b) = 2*(8a - 3b). That seems like a good start.Let me write that out: 7*(6a - 5b) = 2*(8a - 3b). Now, I need to distribute the 7 and the 2 into the parentheses. Let me do that carefully.On the left side, 7 times 6a is 42a, and 7 times -5b is -35b. So, the left side becomes 42a - 35b. On the right side, 2 times 8a is 16a, and 2 times -3b is -6b. So, the right side becomes 16a - 6b.Now, my equation looks like this: 42a - 35b = 16a - 6b. I need to get all the a terms on one side and the b terms on the other. Let me subtract 16a from both sides to move the a terms to the left. That would give me 42a - 16a - 35b = -6b. Simplifying 42a - 16a gives me 26a, so now I have 26a - 35b = -6b.Next, I want to get all the b terms on the right side. So, I'll add 35b to both sides. That gives me 26a = -6b + 35b. Simplifying the right side, -6b + 35b is 29b. So now, the equation is 26a = 29b.I need the ratio of a to b, which is a/b. To find that, I can divide both sides of the equation by b and then divide both sides by 26. So, dividing both sides by b gives me 26a/b = 29. Then, dividing both sides by 26, I get a/b = 29/26.Wait, let me double-check that. Starting from 26a = 29b, if I divide both sides by b, I get 26a/b = 29. Then, dividing both sides by 26, I indeed get a/b = 29/26. That seems correct.Just to make sure I didn't make any mistakes in my earlier steps, let me go through them again. I started with (6a - 5b)/(8a - 3b) = 2/7, cross-multiplied to get 7*(6a - 5b) = 2*(8a - 3b). Expanding both sides gave me 42a - 35b = 16a - 6b. Subtracting 16a from both sides: 26a - 35b = -6b. Adding 35b to both sides: 26a = 29b. Dividing both sides by b and then by 26: a/b = 29/26.Everything checks out. So, the ratio of a to b is 29/26.**Final Answer**The ratio of ( a ) to ( b ) is boxed{dfrac{29}{26}}.

💡Alright, so I have this problem here where I need to figure out the relationship between three values: a, b, and c. They’re defined as follows:- ( a = ln{0.5} )- ( b = pi^{0.4} )- ( c = 0.3^{pi} )And the question is asking me to determine the order of these three values from smallest to largest. The options given are:A: ( a < b < c )B: ( c < a < b )C: ( a < c < b )D: ( b < c < a )Hmm, okay. Let me think about each of these one by one.Starting with ( a = ln{0.5} ). I know that the natural logarithm of a number less than 1 is negative because the logarithm function is only defined for positive numbers, and it decreases as the input approaches zero. Since 0.5 is less than 1, ( ln{0.5} ) should be negative. Specifically, I remember that ( ln{1} = 0 ), so ( ln{0.5} ) must be less than zero. Maybe around -0.693? I think that's correct because ( e^{-0.693} ) is approximately 0.5. So, ( a ) is a negative number.Next, ( b = pi^{0.4} ). Pi is approximately 3.1416, and raising it to the power of 0.4. Since any number greater than 1 raised to a positive power will be greater than 1, right? So, ( pi^{0.4} ) should be greater than 1. Let me see, 3.1416 to the power of 0.4. Maybe I can approximate it. I know that ( pi^{0.5} ) is about 1.772, so 0.4 is a bit less than 0.5, so maybe around 1.5 or so? Hmm, not too sure, but definitely greater than 1.Then, ( c = 0.3^{pi} ). This is a number between 0 and 1 raised to the power of pi, which is approximately 3.1416. When you raise a number between 0 and 1 to a positive exponent, the result is smaller than the original number. So, 0.3 raised to the power of pi should be less than 0.3. Since 0.3 is already less than 1, ( c ) will be a small positive number, but definitely less than 1.So, summarizing:- ( a ) is negative.- ( b ) is greater than 1.- ( c ) is between 0 and 1.Therefore, the order from smallest to largest should be ( a < c < b ), which is option C.Wait, let me double-check to make sure I didn't make a mistake. Maybe I should calculate approximate values for each.Starting with ( a = ln{0.5} ). As I thought earlier, ( ln{0.5} ) is approximately -0.6931.For ( b = pi^{0.4} ), let's compute that. Taking natural logarithm first: ( ln{pi} approx 1.1447 ). Then, ( ln{b} = 0.4 times ln{pi} approx 0.4 times 1.1447 approx 0.4579 ). So, ( b = e^{0.4579} approx 1.58 ). Okay, so approximately 1.58, which is indeed greater than 1.For ( c = 0.3^{pi} ), again, taking natural logarithm: ( ln{c} = pi times ln{0.3} approx 3.1416 times (-1.2039) approx -3.783 ). Therefore, ( c = e^{-3.783} approx 0.0237 ). So, approximately 0.0237, which is indeed between 0 and 1, and much smaller than 0.3.So, plugging in the approximate values:- ( a approx -0.6931 )- ( c approx 0.0237 )- ( b approx 1.58 )So, the order is ( a < c < b ), which confirms that option C is correct.I think that's solid. I don't see any mistakes in my reasoning. So, yeah, I'm confident that the answer is C.**Final Answer**boxed{C}

💡Okay, so I have this problem about a hyperbola, and I need to find its eccentricity. Let me try to break it down step by step. First, the equation of the hyperbola is given as (frac{x^2}{a^2} - frac{y^2}{b^2} = 1). I remember that for hyperbolas, the standard form is similar to this, and the foci are located at ((pm c, 0)), where (c^2 = a^2 + b^2). So, the foci (F_1) and (F_2) are at ((-c, 0)) and ((c, 0)) respectively.Next, the problem says that a line perpendicular to the x-axis passes through (F_2) and intersects the hyperbola at points (A) and (B). A line perpendicular to the x-axis is a vertical line, so its equation must be (x = c) because it passes through (F_2(c, 0)). To find the points (A) and (B), I need to substitute (x = c) into the hyperbola equation:[frac{c^2}{a^2} - frac{y^2}{b^2} = 1]Let me solve for (y^2):[frac{y^2}{b^2} = frac{c^2}{a^2} - 1][y^2 = b^2 left( frac{c^2}{a^2} - 1 right)]Hmm, I can factor this a bit more:[y^2 = frac{b^2 c^2}{a^2} - b^2 = frac{b^2 (c^2 - a^2)}{a^2}]But wait, since (c^2 = a^2 + b^2), substituting that in:[y^2 = frac{b^2 (a^2 + b^2 - a^2)}{a^2} = frac{b^4}{a^2}]So, (y = pm frac{b^2}{a}). Therefore, the points (A) and (B) are ((c, frac{b^2}{a})) and ((c, -frac{b^2}{a})) respectively.Now, the centroid (G) of triangle (ABF_1). The centroid is the average of the coordinates of the three vertices. So, let's find the coordinates of (G).First, the coordinates of the three points:- (A(c, frac{b^2}{a}))- (B(c, -frac{b^2}{a}))- (F_1(-c, 0))So, the centroid (G) will have coordinates:[x_G = frac{c + c + (-c)}{3} = frac{c}{3}][y_G = frac{frac{b^2}{a} + (-frac{b^2}{a}) + 0}{3} = 0]So, (G) is at ((frac{c}{3}, 0)).Next, the problem states that (overrightarrow{GA} cdot overrightarrow{F_1B} = 0). That means the vectors (GA) and (F_1B) are perpendicular to each other.Let me find the vectors (GA) and (F_1B).First, vector (GA) is from (G) to (A):[overrightarrow{GA} = (c - frac{c}{3}, frac{b^2}{a} - 0) = (frac{2c}{3}, frac{b^2}{a})]Vector (F_1B) is from (F_1) to (B):[overrightarrow{F_1B} = (c - (-c), -frac{b^2}{a} - 0) = (2c, -frac{b^2}{a})]Now, the dot product of these two vectors should be zero:[overrightarrow{GA} cdot overrightarrow{F_1B} = left( frac{2c}{3} times 2c right) + left( frac{b^2}{a} times -frac{b^2}{a} right) = 0]Calculating each term:First term: (frac{2c}{3} times 2c = frac{4c^2}{3})Second term: (frac{b^2}{a} times -frac{b^2}{a} = -frac{b^4}{a^2})So, putting it together:[frac{4c^2}{3} - frac{b^4}{a^2} = 0]Let me rearrange this equation:[frac{4c^2}{3} = frac{b^4}{a^2}]Multiply both sides by (3a^2):[4c^2 a^2 = 3b^4]Hmm, I can write this as:[4a^2 c^2 = 3b^4]I know that (c^2 = a^2 + b^2), so let's substitute that in:[4a^2 (a^2 + b^2) = 3b^4][4a^4 + 4a^2 b^2 = 3b^4]Let me bring all terms to one side:[4a^4 + 4a^2 b^2 - 3b^4 = 0]This looks a bit complicated. Maybe I can express everything in terms of (e), the eccentricity, since that's what we're trying to find. I remember that (e = frac{c}{a}), and since (c^2 = a^2 + b^2), we can write (b^2 = c^2 - a^2 = a^2(e^2 - 1)).Let me substitute (b^2 = a^2(e^2 - 1)) into the equation:First, compute (b^4):[b^4 = (a^2(e^2 - 1))^2 = a^4(e^2 - 1)^2]Now, substitute into the equation:[4a^4 + 4a^2 times a^2(e^2 - 1) - 3a^4(e^2 - 1)^2 = 0]Simplify each term:First term: (4a^4)Second term: (4a^4(e^2 - 1))Third term: (-3a^4(e^2 - 1)^2)Factor out (a^4):[a^4 left[4 + 4(e^2 - 1) - 3(e^2 - 1)^2 right] = 0]Since (a^4) is not zero, we can focus on the expression inside the brackets:[4 + 4(e^2 - 1) - 3(e^2 - 1)^2 = 0]Let me let (u = e^2 - 1) to simplify the equation:[4 + 4u - 3u^2 = 0]Rewriting:[-3u^2 + 4u + 4 = 0][3u^2 - 4u - 4 = 0]Now, solve for (u) using the quadratic formula:[u = frac{4 pm sqrt{(-4)^2 - 4 times 3 times (-4)}}{2 times 3}][u = frac{4 pm sqrt{16 + 48}}{6}][u = frac{4 pm sqrt{64}}{6}][u = frac{4 pm 8}{6}]So, two solutions:1. (u = frac{4 + 8}{6} = frac{12}{6} = 2)2. (u = frac{4 - 8}{6} = frac{-4}{6} = -frac{2}{3})But (u = e^2 - 1), and since (e > 1) for hyperbolas, (u) must be positive. So, (u = 2).Thus,[e^2 - 1 = 2][e^2 = 3][e = sqrt{3}]So, the eccentricity of the hyperbola is (sqrt{3}).Wait, let me double-check my steps to make sure I didn't make a mistake.1. Found points (A) and (B) correctly by plugging (x = c) into the hyperbola equation.2. Calculated centroid (G) correctly by averaging the coordinates.3. Found vectors (GA) and (F_1B) correctly.4. Set up the dot product equation correctly and solved for (c^2) and (b^4).5. Substituted (c^2 = a^2 + b^2) correctly.6. Expressed (b^2) in terms of (a^2) and (e), leading to an equation in terms of (e).7. Solved the quadratic equation correctly, discarding the negative solution since (e > 1).Everything seems to check out. So, I think the eccentricity is indeed (sqrt{3}).**Final Answer**The eccentricity of the hyperbola is (boxed{sqrt{3}}).

💡Alright, so I have this problem here: Compute the tangent of the sum of two arctangent functions. Specifically, it's tan(arctan(3/5) + arctan(2/3)). Hmm, okay, let me try to figure this out step by step.First off, I remember that arctan(x) gives me an angle whose tangent is x. So, arctan(3/5) is an angle whose tangent is 3/5, and arctan(2/3) is another angle whose tangent is 2/3. Let me call these two angles something, maybe α and β. So, α = arctan(3/5) and β = arctan(2/3). That means tan(α) = 3/5 and tan(β) = 2/3.Now, the problem is asking for tan(α + β). I think there's a formula for the tangent of the sum of two angles. Let me recall... I think it's tan(α + β) = (tan α + tan β) / (1 - tan α tan β). Yeah, that sounds right. So, I can use this formula to find the tangent of the sum of these two angles.Okay, so plugging in the values I have: tan(α + β) = (3/5 + 2/3) / (1 - (3/5)(2/3)). Let me compute the numerator and the denominator separately to avoid mistakes.Starting with the numerator: 3/5 + 2/3. To add these fractions, I need a common denominator. The denominators are 5 and 3, so the least common denominator is 15. Converting both fractions: 3/5 is equal to 9/15, and 2/3 is equal to 10/15. Adding them together: 9/15 + 10/15 = 19/15. So, the numerator is 19/15.Now, the denominator: 1 - (3/5)(2/3). Let me compute the product first: (3/5)(2/3). Multiplying the numerators: 3*2 = 6, and the denominators: 5*3 = 15. So, that's 6/15. Simplifying that, 6/15 reduces to 2/5. So, now the denominator becomes 1 - 2/5. To subtract these, I can write 1 as 5/5, so 5/5 - 2/5 = 3/5. Therefore, the denominator is 3/5.Putting it all together: tan(α + β) = (19/15) / (3/5). Dividing by a fraction is the same as multiplying by its reciprocal, so this becomes (19/15) * (5/3). Let's compute that: 19*5 = 95, and 15*3 = 45. So, that's 95/45. Simplifying this fraction by dividing numerator and denominator by 5: 95 ÷ 5 = 19, and 45 ÷ 5 = 9. So, it simplifies to 19/9.Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting with the formula: tan(α + β) = (tan α + tan β) / (1 - tan α tan β). Plugging in 3/5 and 2/3, I got 19/15 in the numerator and 3/5 in the denominator. Dividing those gives 19/9. That seems correct.But just to be thorough, let me visualize this. If α is arctan(3/5), then in a right triangle, the opposite side is 3 and the adjacent side is 5. Similarly, for β = arctan(2/3), the opposite side is 2 and the adjacent side is 3. If I imagine adding these two angles, the resulting angle's tangent should be 19/9, which is what I got. That seems reasonable.I wonder if there's another way to approach this problem, maybe by converting arctan to another function or using some identity. But I think the method I used is straightforward and correct. The tangent addition formula is reliable here because both angles are positive and less than 90 degrees, so their sum is less than 180 degrees, and the tangent is defined.Another thought: maybe I can use the sine and cosine addition formulas instead. Let's see. If I compute sin(α + β) and cos(α + β), then tan(α + β) would be sin(α + β)/cos(α + β). But that might involve more steps since I'd have to find sin α, cos α, sin β, and cos β first.Let me try that approach just for verification. Starting with α = arctan(3/5). So, in a right triangle, opposite = 3, adjacent = 5, hypotenuse = sqrt(3² + 5²) = sqrt(9 + 25) = sqrt(34). Therefore, sin α = 3/sqrt(34) and cos α = 5/sqrt(34).Similarly, for β = arctan(2/3), opposite = 2, adjacent = 3, hypotenuse = sqrt(2² + 3²) = sqrt(4 + 9) = sqrt(13). So, sin β = 2/sqrt(13) and cos β = 3/sqrt(13).Now, using the sine addition formula: sin(α + β) = sin α cos β + cos α sin β. Plugging in the values: (3/sqrt(34))(3/sqrt(13)) + (5/sqrt(34))(2/sqrt(13)). That's (9)/(sqrt(34)*sqrt(13)) + (10)/(sqrt(34)*sqrt(13)) = (19)/(sqrt(34)*sqrt(13)).Similarly, the cosine addition formula: cos(α + β) = cos α cos β - sin α sin β. Plugging in the values: (5/sqrt(34))(3/sqrt(13)) - (3/sqrt(34))(2/sqrt(13)) = (15)/(sqrt(34)*sqrt(13)) - (6)/(sqrt(34)*sqrt(13)) = (9)/(sqrt(34)*sqrt(13)).So, tan(α + β) = sin(α + β)/cos(α + β) = (19)/(sqrt(34)*sqrt(13)) divided by (9)/(sqrt(34)*sqrt(13)). The denominators cancel out, leaving 19/9. So, that's the same result as before. Good, so both methods give me the same answer, which is reassuring.I think I'm confident now that 19/9 is the correct value for tan(arctan(3/5) + arctan(2/3)). It's always nice to verify using different methods to make sure I didn't make any calculation errors.Just to recap, the key steps were:1. Recognize that arctan(3/5) and arctan(2/3) are angles with known tangents.2. Use the tangent addition formula to compute the tangent of their sum.3. Simplify the resulting expression carefully, making sure to find a common denominator and simplify fractions.4. Optionally, verify using the sine and cosine addition formulas to ensure consistency.This problem was a good exercise in applying trigonometric identities and fraction operations. It's easy to make a mistake with the signs or the fractions, so taking it step by step and double-checking each part helped prevent errors.I also thought about whether the angles could be in different quadrants, but since both 3/5 and 2/3 are positive, both arctan values are in the first quadrant, so their sum is also in the first or second quadrant. However, since the tangent is positive (19/9), the angle must be in the first quadrant, which aligns with our calculation.Another thing I considered was whether the sum of the angles could be more than 90 degrees, which would affect the tangent value. But since 3/5 is less than 1 and 2/3 is also less than 1, their arctans are both less than 45 degrees, so their sum is less than 90 degrees. Therefore, the tangent is positive, which matches our result.In summary, by methodically applying the tangent addition formula and verifying with an alternative approach, I arrived at the solution of 19/9. It's a solid answer, and I feel confident about it after checking my work.

💡Okay, so I have this system of equations to solve:begin{cases}x + (m + 1)y + m - 2 = 0 2mx + 4y + 16 = 0end{cases}Hmm, I need to solve for x and y in terms of m and also discuss the possible cases for the solutions. Alright, let's start by understanding what this system is about.First, I remember that when solving a system of linear equations, there are a few methods I can use, like substitution, elimination, or using matrices. Since the problem mentions discussing possible cases, maybe I should consider using matrices because determinants can tell me about the nature of the solutions.So, let's write the system in matrix form. The coefficient matrix D would be:D = begin{vmatrix}1 & m + 1 2m & 4end{vmatrix}To find the determinant D, I calculate (1)(4) - (m + 1)(2m) = 4 - 2m(m + 1). Let me expand that: 4 - 2m^2 - 2m. So, D = 4 - 2m^2 - 2m.Now, if D is not zero, the system has a unique solution. If D is zero, then the system might have either infinitely many solutions or no solution, depending on the constants on the right side of the equations.Let me write down the determinant again:D = 4 - 2m^2 - 2m.I can factor this to see when it's zero. Let's factor out a -2:D = -2(m^2 + m - 2).Now, factor the quadratic inside the parentheses:m^2 + m - 2 = (m + 2)(m - 1).So, D = -2(m + 2)(m - 1). Therefore, D = 0 when m = -2 or m = 1.So, for m ≠ -2 and m ≠ 1, D ≠ 0, and the system has a unique solution. For m = -2 and m = 1, D = 0, so we need to check if the system is consistent or inconsistent in those cases.Alright, now let's find the unique solution when D ≠ 0. Using Cramer's Rule, I can find x and y.First, for x:Replace the first column of D with the constants from the equations. The constants are (m - 2) and 16, right?So, the matrix for x is:D_x = begin{vmatrix}m - 2 & m + 1 16 & 4end{vmatrix}Calculating this determinant: (m - 2)(4) - (m + 1)(16) = 4m - 8 - 16m - 16 = -12m - 24.So, x = D_x / D = (-12m - 24) / (4 - 2m^2 - 2m).Wait, I can factor numerator and denominator:Numerator: -12(m + 2)Denominator: -2(m + 2)(m - 1)So, x = [-12(m + 2)] / [-2(m + 2)(m - 1)] = [12(m + 2)] / [2(m + 2)(m - 1)] = 6 / (m - 1).Wait, but in the original calculation, I had D = 4 - 2m^2 - 2m, which is equal to -2(m^2 + m - 2) = -2(m + 2)(m - 1). So, when I write x, I can cancel out (m + 2) from numerator and denominator, but only when m ≠ -2, which is already considered because we're in the case where D ≠ 0, so m ≠ -2 and m ≠ 1.So, x = 6 / (m - 1). Hmm, but in the previous step, I had x = 6 / (1 - m). Wait, that's just a sign difference. Let me check my calculation again.Wait, D_x was (-12m - 24) = -12(m + 2). D was 4 - 2m^2 - 2m = -2(m + 2)(m - 1). So, x = (-12(m + 2)) / (-2(m + 2)(m - 1)) = (12(m + 2)) / (2(m + 2)(m - 1)) = 6 / (m - 1). So, x = 6 / (m - 1).But in the initial answer, it was written as 6 / (1 - m). That's just the negative of this. So, perhaps I made a sign error. Let me double-check.Wait, in the determinant D_x, I had:(m - 2)(4) - (m + 1)(16) = 4m - 8 - 16m - 16 = -12m - 24. That's correct.And D is 4 - 2m^2 - 2m = -2m^2 - 2m + 4 = -2(m^2 + m - 2) = -2(m + 2)(m - 1). Correct.So, x = (-12m - 24) / (-2m^2 - 2m + 4) = (-12(m + 2)) / (-2(m + 2)(m - 1)) = [12(m + 2)] / [2(m + 2)(m - 1)] = 6 / (m - 1). So, x = 6 / (m - 1).But in the initial answer, it was written as 6 / (1 - m). Wait, that's the same as -6 / (m - 1). So, perhaps I made a mistake in the sign somewhere.Wait, let's go back to D_x:D_x = |(m - 2) (m + 1); 16 4| = (m - 2)(4) - (m + 1)(16) = 4m - 8 - 16m - 16 = -12m -24. Correct.D = |1 (m + 1); 2m 4| = 4 - 2m(m + 1) = 4 - 2m^2 - 2m. Correct.So, x = (-12m -24)/(4 - 2m^2 - 2m) = (-12(m + 2))/(-2(m + 2)(m - 1)) = [12(m + 2)]/[2(m + 2)(m - 1)] = 6/(m - 1). So, x = 6/(m - 1).But in the initial answer, it was written as 6/(1 - m). So, perhaps I should write it as -6/(m - 1) = 6/(1 - m). So, yes, x = 6/(1 - m). That makes sense because 6/(1 - m) is the same as -6/(m - 1). So, both are correct, just written differently.Now, let's find y using Cramer's Rule.For y, replace the second column of D with the constants.So, the matrix for y is:D_y = begin{vmatrix}1 & m - 2 2m & 16end{vmatrix}Calculating this determinant: (1)(16) - (m - 2)(2m) = 16 - 2m(m - 2) = 16 - 2m^2 + 4m.So, y = D_y / D = (16 - 2m^2 + 4m) / (4 - 2m^2 - 2m).Let me factor numerator and denominator:Numerator: -2m^2 + 4m + 16 = -2(m^2 - 2m - 8) = -2(m - 4)(m + 2).Denominator: -2(m + 2)(m - 1).So, y = [-2(m - 4)(m + 2)] / [-2(m + 2)(m - 1)] = (m - 4)/(m - 1).Again, since m ≠ -2 and m ≠ 1, we can cancel out (m + 2) and the -2.So, y = (m - 4)/(m - 1). Alternatively, we can write it as (4 - m)/(1 - m) if we factor out a negative from numerator and denominator.So, y = (4 - m)/(1 - m) = (m - 4)/(m - 1). Both are correct.So, summarizing, when m ≠ 1 and m ≠ -2, the system has a unique solution:x = 6/(1 - m)y = (m - 4)/(1 - m)Now, let's check the cases when m = 1 and m = -2.First, m = 1.Substitute m = 1 into the original equations:First equation: x + (1 + 1)y + 1 - 2 = 0 → x + 2y - 1 = 0.Second equation: 2(1)x + 4y + 16 = 0 → 2x + 4y + 16 = 0.So, the system becomes:x + 2y = 12x + 4y = -16Let me write them as:1) x + 2y = 12) 2x + 4y = -16If I multiply equation 1 by 2, I get 2x + 4y = 2.But equation 2 is 2x + 4y = -16.So, 2x + 4y cannot be both 2 and -16. Therefore, the system is inconsistent, meaning no solution when m = 1.Now, m = -2.Substitute m = -2 into the original equations:First equation: x + (-2 + 1)y + (-2) - 2 = 0 → x - y - 4 = 0.Second equation: 2(-2)x + 4y + 16 = 0 → -4x + 4y + 16 = 0.So, the system becomes:x - y = 4-4x + 4y = -16Let me write them as:1) x - y = 42) -4x + 4y = -16If I multiply equation 1 by 4, I get 4x - 4y = 16.Now, add this to equation 2:4x - 4y -4x + 4y = 16 -16 → 0 = 0.Hmm, that's an identity, which suggests that the two equations are dependent. Wait, but let me check.Wait, equation 1: x - y = 4Equation 2: -4x + 4y = -16If I factor equation 2: -4(x - y) = -16 → x - y = 4.Which is the same as equation 1. So, both equations are the same, meaning infinitely many solutions.Wait, but in the initial answer, it was said that when m = -2, the system has no solution. That contradicts my current conclusion.Wait, let me check again.Original equations when m = -2:First equation: x + (-2 + 1)y + (-2) - 2 = 0 → x - y -4 = 0.Second equation: 2*(-2)x + 4y +16 = 0 → -4x + 4y +16 = 0.So, equation 1: x - y = 4Equation 2: -4x + 4y = -16Let me simplify equation 2: divide both sides by -4: x - y = 4.So, both equations are identical. Therefore, the system has infinitely many solutions, not no solution.Wait, that contradicts the initial answer. So, perhaps I made a mistake in the initial analysis.Wait, let me double-check.When m = -2:First equation: x + (-2 + 1)y + (-2) - 2 = x - y -4 = 0 → x - y = 4.Second equation: 2*(-2)x + 4y +16 = -4x +4y +16 =0 → -4x +4y = -16 → divide by -4: x - y =4.So, both equations are the same, meaning infinitely many solutions.But in the initial answer, it was stated that when m = -2, the system has no solution. That must be incorrect.Similarly, when m =1, the system is inconsistent, as we saw.So, perhaps the initial answer had a mistake in the case of m = -2.Wait, let me check the initial answer again.In the initial answer, it was written:"When m = -2, the original system becomes:x - y -4 =0-4x +4y +16=0which has no solution."But as we saw, these two equations are multiples of each other, so they are dependent and have infinitely many solutions.Therefore, the initial answer was incorrect in stating that m = -2 leads to no solution. Instead, it leads to infinitely many solutions.Similarly, when m =1, the system is inconsistent, so no solution.Therefore, the correct conclusion is:- For m ≠1 and m ≠-2, unique solution.- For m =1, no solution.- For m =-2, infinitely many solutions.So, the initial answer had an error in the case of m =-2.Therefore, in the final answer, I should correct that.So, to summarize:When m ≠1 and m ≠-2, the system has a unique solution:x = 6/(1 - m)y = (m -4)/(1 - m)When m =1, the system is inconsistent, so no solution.When m =-2, the system has infinitely many solutions, as the two equations are dependent.Therefore, the possible cases are:- Unique solution when m ≠1 and m ≠-2.- No solution when m =1.- Infinitely many solutions when m =-2.So, the initial answer had an error in the case of m =-2, stating no solution instead of infinitely many solutions.Therefore, the corrected conclusion is as above.

💡Okay, so I have this problem where I need to find the minimum value of the expression:[frac{a^2 + b^2}{a + b} + frac{a^2 + c^2}{a + c} + frac{b^2 + c^2}{b + c}]given that (a), (b), and (c) are positive real numbers adding up to 9. Hmm, let me think about how to approach this.First, I remember that when dealing with expressions like (frac{x^2 + y^2}{x + y}), there might be some inequality that can help simplify or bound this term. Maybe the Cauchy-Schwarz inequality or the AM-GM inequality? Let me recall.Wait, the QM-AM inequality! That's the Quadratic Mean - Arithmetic Mean inequality. It states that for non-negative real numbers, the quadratic mean is always greater than or equal to the arithmetic mean. The formula is:[sqrt{frac{x^2 + y^2}{2}} geq frac{x + y}{2}]If I square both sides to get rid of the square root, it becomes:[frac{x^2 + y^2}{2} geq left(frac{x + y}{2}right)^2]Multiplying both sides by 2 gives:[x^2 + y^2 geq frac{(x + y)^2}{2}]So, if I divide both sides by (x + y), assuming (x + y neq 0), which it isn't since (a), (b), and (c) are positive, I get:[frac{x^2 + y^2}{x + y} geq frac{x + y}{2}]Ah, that's useful! So, applying this to each term in the original expression:1. For (frac{a^2 + b^2}{a + b}), we have (frac{a^2 + b^2}{a + b} geq frac{a + b}{2}).2. Similarly, (frac{a^2 + c^2}{a + c} geq frac{a + c}{2}).3. And (frac{b^2 + c^2}{b + c} geq frac{b + c}{2}).Now, if I add all these inequalities together:[frac{a^2 + b^2}{a + b} + frac{a^2 + c^2}{a + c} + frac{b^2 + c^2}{b + c} geq frac{a + b}{2} + frac{a + c}{2} + frac{b + c}{2}]Let me compute the right-hand side:[frac{a + b}{2} + frac{a + c}{2} + frac{b + c}{2} = frac{2a + 2b + 2c}{2} = a + b + c]But we know from the problem statement that (a + b + c = 9). So, the right-hand side simplifies to 9.Therefore, the original expression is greater than or equal to 9. Now, the question is: when does equality hold? Because if I can find values of (a), (b), and (c) that make the inequality an equality, then 9 is indeed the minimum.Looking back at the QM-AM inequality, equality holds when all the numbers are equal. So, in each case:- For (frac{a^2 + b^2}{a + b} = frac{a + b}{2}), we need (a = b).- Similarly, (frac{a^2 + c^2}{a + c} = frac{a + c}{2}) implies (a = c).- And (frac{b^2 + c^2}{b + c} = frac{b + c}{2}) implies (b = c).So, all three variables must be equal. That is, (a = b = c). Since (a + b + c = 9), each of them must be 3.Let me verify this. If (a = b = c = 3), then each term in the original expression becomes:[frac{3^2 + 3^2}{3 + 3} = frac{9 + 9}{6} = frac{18}{6} = 3]So, adding all three terms: (3 + 3 + 3 = 9), which matches our earlier result.Therefore, the minimum value of the expression is indeed 9, achieved when (a = b = c = 3).I think this makes sense. I don't see any mistakes in the reasoning. The key was recognizing the application of the QM-AM inequality to each fraction, which allowed me to bound the expression from below and then find when equality occurs.**Final Answer**The minimum value is boxed{9}.