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question:Determine the values of A and B such that frac{3x+5}{x^2-x-42}=frac{A}{x-7}+frac{B}{x+6}. Express your answer in the form (A,B).
answer:First, we factor the denominator of the left side of the equation as (x-7)(x+6). We can then rewrite the equation as frac{3x+5}{(x-7)(x+6)}=frac{A}{x-7}+frac{B}{x+6}. To combine the fractions on the right side, we need a common denominator, which is (x-7)(x+6). Multiplying the first fraction on the right side by frac{x+6}{x+6} and the second fraction by frac{x-7}{x-7}, we have frac{3x+5}{(x-7)(x+6)}=frac{A(x+6)}{(x-7)(x+6)}+frac{B(x-7)}{(x-7)(x+6)}. Combining the fractions on the right side, we have frac{3x+5}{(x-7)(x+6)}=frac{Ax+6A+Bx-7B}{(x-7)(x+6)}. The numerators on both sides must be equal, so we have 3x+5=Ax+6A+Bx-7B. Matching the coefficients of the x terms and constant terms on both sides, we have A+B=3 and 6A-7B=5. Solving this system of equations, we find A=2 and B=1. Therefore, the values of A and B are (A,B)=boxed{(2,1)}.The answer is: (2,1)
question:f a @ b is defined as a @ b = 3a - 3b, what is the value of 3 @ 5?
answer:To find the value of 3 @ 5, we substitute a = 3 and b = 5 into the expression 3a - 3b. So, 3 @ 5 = 3(3) - 3(5) = 9 - 15 = boxed{-6}. The answer is: -6
question:ggy is training for a marathon. On Monday, he runs 3 miles. On Tuesday, he runs 4 miles. On Wednesday, he runs x miles. On Thursday, he runs 8 miles. On Friday, he runs 3 miles. Iggy runs at a pace of 1 mile in 10 minutes. What is the total number of hours that Iggy spends running from Monday to Friday? If we know the answer to the above question is 4, what is the value of unknown variable x?
answer:On Monday, Iggy runs 3 miles, which is equivalent to 3 * 10 = 30 minutes. On Tuesday, he runs 4 miles, which is equivalent to 4 * 10 = 40 minutes. On Wednesday, he runs x miles, which is equivalent to x * 10 minutes. On Thursday, he runs 8 miles, which is equivalent to 8 * 10 = 80 minutes. On Friday, he runs 3 miles, which is equivalent to 3 * 10 = 30 minutes. The total time Iggy spends running from Monday to Friday is: 30 + 40 + (x * 10) + 80 + 30. We are given that the total time is 4 hours, or 240 minutes, so we can write: 30 + 40 + (x * 10) + 80 + 30 = 240. Simplifying the left side, we get: 180 + (x * 10) = 240. Subtracting 180 from both sides, we get: x * 10 = 60. Dividing both sides by 10, we get: x = 6. The value of x is 6. 6 The answer is: 6
question:f John bakes 6 dozen cookies for a bake sale, selling each cookie for 1.5 and incurring a cost of 0.25 per cookie, and he splits the profit equally between two charities, how much will each charity receive?
answer:John bakes 6 dozen cookies, which is 6 x 12 = 72 cookies. If he sells each cookie for 1.5, his total revenue will be 72 x 1.5 = 108. His cost per cookie is 0.25, so his total cost will be 72 x 0.25 = 18. Therefore, his total profit will be 108 - 18 = 90. Since he splits the profit equally between two charities, each charity will receive 90 / 2 = 45. Each charity will receive 45. 45 The answer is: 45