💡Okay, so I have this equation here: |x - 4| + |5 + y| = 0. I need to find the value of 2x + 3y. Hmm, let me think about how to approach this.First, I remember that absolute values are always non-negative. That means |x - 4| is greater than or equal to zero, and |5 + y| is also greater than or equal to zero. Since their sum is zero, each of them must individually be zero. Because if one of them was positive, the other would have to be negative to make the sum zero, but absolute values can't be negative. So, both |x - 4| and |5 + y| must equal zero.Alright, so if |x - 4| = 0, then x - 4 must be zero. That means x = 4. Similarly, if |5 + y| = 0, then 5 + y must be zero. So, y = -5.Now that I have the values of x and y, I can plug them into the expression 2x + 3y. Let's do that step by step.First, calculate 2x: 2 times x, which is 2 times 4. That gives me 8.Next, calculate 3y: 3 times y, which is 3 times (-5). That gives me -15.Now, add those two results together: 8 + (-15). Adding a negative is the same as subtracting, so 8 - 15 equals -7.Wait, let me double-check my calculations to make sure I didn't make a mistake. So, x = 4 and y = -5. Plugging into 2x + 3y: 2*4 is 8, and 3*(-5) is -15. Adding them together: 8 + (-15) is indeed -7. Okay, that seems correct.I think I'm confident with this answer. The value of 2x + 3y is -7.

💡First, I recognize that Xiaoting has taken five math tests. The average score is 85, so the total sum of all five test scores is ( 85 times 5 = 425 ).The median score is 86, which means that when the scores are arranged in ascending order, the middle score is 86. This implies that two scores are below 86 and two scores are above 86.The mode is 88, indicating that 88 appears more frequently than any other score. Since there are five tests, and the mode is 88, it must appear at least twice. Therefore, there are two scores of 88.With the median being 86 and the mode being 88, the three known scores are 86, 88, and 88. Adding these together gives ( 86 + 88 + 88 = 262 ).To find the sum of the two lowest scores, I subtract the sum of the known scores from the total sum: ( 425 - 262 = 163 ).Thus, the sum of the scores of the two lowest tests is 163.

💡First, I need to determine the total amount of solution Dr. Igor has by multiplying the number of beakers by the amount of solution in each beaker.Next, I'll find out how much solution was originally in each test tube by dividing the total amount of solution by the number of test tubes.By following these steps, I can calculate the amount of solution in each test tube.

💡First, I need to calculate the difference between the two complex numbers ( z_1 = -3 + i ) and ( z_2 = 1 - i ). Subtracting ( z_2 ) from ( z_1 ) involves subtracting their real parts and their imaginary parts separately.So, ( z = z_1 - z_2 = (-3 - 1) + (i - (-i)) = -4 + 2i ).The resulting complex number ( z = -4 + 2i ) corresponds to the point ((-4, 2)) in the complex plane.In the complex plane, the quadrants are determined by the signs of the real and imaginary parts. A negative real part and a positive imaginary part place the point in the second quadrant.

💡Okay, so I have this problem here: If ( g(3x - 4) = 4x + 6 ) for all real ( x ), then what is ( g(2) )? Hmm, I need to figure out what ( g(2) ) is. Let me think about how to approach this.First, I know that ( g ) is a function, and it's defined in terms of ( 3x - 4 ). So, ( g ) takes an input, which is ( 3x - 4 ), and gives an output of ( 4x + 6 ). But I need to find ( g(2) ), which means I need to find the value of ( x ) such that ( 3x - 4 = 2 ). That way, when I plug that ( x ) into the equation, I'll get ( g(2) ).So, let me set up the equation ( 3x - 4 = 2 ) and solve for ( x ). Adding 4 to both sides gives me ( 3x = 6 ). Then, dividing both sides by 3, I find that ( x = 2 ). Okay, so when ( x = 2 ), the input to the function ( g ) is 2.Now, I need to find ( g(2) ), which is the output when the input is 2. But according to the given equation, ( g(3x - 4) = 4x + 6 ). Since I've found that ( x = 2 ) makes the input 2, I can substitute ( x = 2 ) into the right side of the equation to find ( g(2) ).Substituting ( x = 2 ) into ( 4x + 6 ), I get ( 4*2 + 6 = 8 + 6 = 14 ). So, ( g(2) = 14 ).Wait, let me double-check that to make sure I didn't make any mistakes. If ( x = 2 ), then ( 3x - 4 = 3*2 - 4 = 6 - 4 = 2 ), which matches the input we wanted. And substituting ( x = 2 ) into ( 4x + 6 ) gives ( 8 + 6 = 14 ), which seems correct.Is there another way to approach this problem? Maybe by expressing ( g ) as a function of ( y ), where ( y = 3x - 4 ). Let's try that.Let ( y = 3x - 4 ). Then, solving for ( x ), I get ( x = (y + 4)/3 ). Now, substitute this expression for ( x ) back into the equation ( g(y) = 4x + 6 ). So, ( g(y) = 4*(y + 4)/3 + 6 ).Simplifying that, ( g(y) = (4y + 16)/3 + 6 ). To combine the terms, I can write 6 as ( 18/3 ), so ( g(y) = (4y + 16 + 18)/3 = (4y + 34)/3 ).Now, if I want ( g(2) ), I can substitute ( y = 2 ) into this expression: ( g(2) = (4*2 + 34)/3 = (8 + 34)/3 = 42/3 = 14 ). Yep, that matches my earlier result.So, both methods give me ( g(2) = 14 ). I think that's the correct answer. I don't see any mistakes in my calculations, and both approaches lead to the same result, which makes me more confident that it's right.Another way to think about it is to recognize that ( g ) is a linear function because both sides of the equation are linear in ( x ). So, ( g ) can be expressed as ( g(z) = az + b ), where ( a ) and ( b ) are constants. Let's try to find ( a ) and ( b ).Given ( g(3x - 4) = 4x + 6 ), and assuming ( g(z) = az + b ), then ( g(3x - 4) = a*(3x - 4) + b = 3a x - 4a + b ). This should equal ( 4x + 6 ).So, setting the coefficients equal, we have:- Coefficient of ( x ): ( 3a = 4 ) → ( a = 4/3 )- Constant term: ( -4a + b = 6 ). Substituting ( a = 4/3 ), we get ( -4*(4/3) + b = 6 ) → ( -16/3 + b = 6 ) → ( b = 6 + 16/3 = 18/3 + 16/3 = 34/3 )Therefore, ( g(z) = (4/3)z + 34/3 ). Now, plugging in ( z = 2 ), we get ( g(2) = (4/3)*2 + 34/3 = 8/3 + 34/3 = 42/3 = 14 ). Again, the same answer.This consistency across different methods makes me confident that ( g(2) = 14 ) is indeed the correct solution.

💡Okay, so I have this problem where I need to find the remainder when a polynomial Q(x) is divided by (x - 15)(x - 19). I know that when Q(x) is divided by x - 15, the remainder is 8, and when it's divided by x - 19, the remainder is 10. Hmm, I remember something about the Remainder Theorem, which says that the remainder when a polynomial is divided by x - a is just Q(a). So, that means Q(15) = 8 and Q(19) = 10.Now, when dividing by a quadratic polynomial like (x - 15)(x - 19), the remainder should be of a lower degree, right? So, it should be a linear polynomial, which I can write as cx + d, where c and d are constants that I need to find.So, I can express Q(x) as:Q(x) = (x - 15)(x - 19)R(x) + cx + dwhere R(x) is the quotient polynomial.Since I know Q(15) = 8, I can plug x = 15 into this equation:Q(15) = (15 - 15)(15 - 19)R(15) + c*15 + dSimplifying, the first term becomes zero because (15 - 15) is zero, so:8 = 15c + dSimilarly, plugging in x = 19:Q(19) = (19 - 15)(19 - 19)R(19) + c*19 + dAgain, the first term is zero because (19 - 19) is zero, so:10 = 19c + dNow I have a system of two equations:1) 15c + d = 82) 19c + d = 10I need to solve for c and d. Let me subtract equation 1 from equation 2 to eliminate d:(19c + d) - (15c + d) = 10 - 819c + d - 15c - d = 24c = 2So, c = 2 / 4 = 1/2Now that I have c, I can plug it back into equation 1 to find d:15*(1/2) + d = 815/2 + d = 8d = 8 - 15/2Converting 8 to halves, that's 16/2, so:d = 16/2 - 15/2 = 1/2So, the remainder when Q(x) is divided by (x - 15)(x - 19) is (1/2)x + 1/2.Wait, let me double-check my calculations. If c is 1/2, then 15c is 15*(1/2) = 7.5, and d is 0.5. So, 7.5 + 0.5 = 8, which matches Q(15) = 8. Similarly, 19c is 19*(1/2) = 9.5, and d is 0.5, so 9.5 + 0.5 = 10, which matches Q(19) = 10. That seems correct.I think I did everything right. I set up the equations based on the Remainder Theorem, solved the system of equations, and verified the solution. So, the remainder is indeed (1/2)x + 1/2.**Final Answer**The remainder when ( Q(x) ) is divided by ( (x - 15)(x - 19) ) is boxed{frac{1}{2}x + frac{1}{2}}.