💡First, I will solve the system of equations to express ( x ) and ( y ) in terms of ( m ).Starting with the two equations:[begin{cases}x + y = 2m + 7 quad text{(1)} x - 2y = 4m - 3 quad text{(2)}end{cases}]Subtracting equation (2) from equation (1):[(x + y) - (x - 2y) = (2m + 7) - (4m - 3)][3y = -2m + 10][y = frac{-2m + 10}{3} quad text{(3)}]Substituting equation (3) back into equation (1):[x + frac{-2m + 10}{3} = 2m + 7][x = 2m + 7 - frac{-2m + 10}{3}][x = frac{8m + 11}{3}]Now, to ensure that both ( x ) and ( y ) are positive integers, the following conditions must be satisfied:[begin{cases}frac{8m + 11}{3} > 0 frac{-2m + 10}{3} > 0end{cases}]Solving these inequalities:[frac{8m + 11}{3} > 0 Rightarrow 8m + 11 > 0 Rightarrow m > -frac{11}{8}][frac{-2m + 10}{3} > 0 Rightarrow -2m + 10 > 0 Rightarrow m < 5]Combining these results, the range of ( m ) is:[-frac{11}{8} < m < 5]The integer values of ( m ) within this range are:[-1, 0, 1, 2, 3, 4]Therefore, the smallest integer value of ( m ) that satisfies the conditions is:[boxed{-1}]

💡Okay, so I need to figure out the domain of this function: f(x) = log base 2 of (3 - x) divided by the square root of (81 - x squared). Hmm, let me break this down step by step.First, I remember that for a function to have a domain, it needs to be defined for all the operations involved. So, in this case, there are two main parts: the logarithm in the numerator and the square root in the denominator. Both of these have specific conditions that need to be satisfied for the function to be defined.Starting with the logarithm part: log base 2 of (3 - x). I know that the argument of a logarithm must be positive because you can't take the log of zero or a negative number. So, I need to set up the inequality:3 - x > 0Solving this, I subtract 3 from both sides:-x > -3Then, I multiply both sides by -1, but I have to remember to flip the inequality sign when I do that:x < 3Okay, so that gives me one condition: x has to be less than 3.Now, moving on to the denominator: square root of (81 - x squared). For a square root to be defined, the expression inside has to be non-negative. Also, since it's in the denominator, it can't be zero because division by zero is undefined. So, I need:81 - x² > 0Let me solve this inequality. I can rewrite it as:x² < 81To solve for x, I take the square root of both sides. Remembering that taking the square root of both sides of an inequality involving squares gives me both the positive and negative roots:|x| < 9Which means:-9 < x < 9So, from the denominator, I get that x has to be between -9 and 9.Now, combining both conditions. From the logarithm, x < 3, and from the denominator, -9 < x < 9. So, the domain of the function is the overlap of these two intervals.Visualizing it on a number line, the denominator restricts x to be between -9 and 9, and the logarithm further restricts it to be less than 3. So, the overlapping interval is from -9 up to, but not including, 3.Therefore, the domain of f(x) is all real numbers x such that -9 < x < 3.I should double-check to make sure I didn't miss anything. Let's see:1. Logarithm: 3 - x > 0 → x < 3. Correct.2. Square root: 81 - x² > 0 → x² < 81 → |x| < 9 → -9 < x < 9. Correct.3. Intersection of both conditions: -9 < x < 3. That seems right.I don't think there are any other restrictions here. The numerator is a logarithm, which we've handled, and the denominator is a square root, which we've also handled. There are no variables in the exponents or anything else that might complicate things further.So, I'm confident that the domain is (-9, 3).

💡First, I need to determine the number of yellow jelly beans in each bag by applying the given percentages to the total number of jelly beans in each bag.For Bag A:32 jelly beans with 40% yellow:32 × 0.40 = 12.8 ≈ 13 yellow jelly beansFor Bag B:34 jelly beans with 30% yellow:34 × 0.30 = 10.2 ≈ 10 yellow jelly beansFor Bag C:36 jelly beans with 25% yellow:36 × 0.25 = 9 yellow jelly beansFor Bag D:38 jelly beans with 15% yellow:38 × 0.15 = 5.7 ≈ 6 yellow jelly beansNext, I'll sum up the yellow jelly beans and the total jelly beans from all bags.Total yellow jelly beans:13 + 10 + 9 + 6 = 38Total jelly beans:32 + 34 + 36 + 38 = 140Now, I'll calculate the percentage of yellow jelly beans in the bowl.Percentage = (38 / 140) × 100% ≈ 27.14%The closest percentage to 27.14% is 27%.

💡Let's denote the number of treats Jane brings as ( T ).1. **Jane's Bread:** - Jane brings 75% as many pieces of bread as treats. - Therefore, Jane's bread = ( 0.75T ).2. **Wanda's Treats:** - Wanda brings half as many treats as Jane. - Therefore, Wanda's treats = ( frac{T}{2} ).3. **Wanda's Bread:** - Wanda brings three times as many pieces of bread as treats. - Therefore, Wanda's bread = ( 3 times frac{T}{2} = 1.5T ).4. **Total Pieces of Bread and Treats:** - The total number of pieces of bread and treats brought by Wanda and Jane is 225. - Therefore, the equation is: [ 0.75T + T + 1.5T + frac{T}{2} = 225 ]5. **Combining Like Terms:** - Combine the terms: [ 0.75T + T + 1.5T + 0.5T = 225 ] [ 3.75T = 225 ]6. **Solving for ( T ):** - Divide both sides by 3.75: [ T = frac{225}{3.75} = 60 ]7. **Finding Wanda's Bread:** - Wanda brought 1.5 times the number of treats she brought. - Wanda's treats = ( frac{60}{2} = 30 ). - Therefore, Wanda's bread = ( 1.5 times 30 = 45 ).

💡First, I need to determine the relative speed at which the two trains are approaching each other. Since they are moving in opposite directions, I add their speeds together.Next, I convert the relative speed from kilometers per hour to meters per second to match the time given in seconds.Finally, I calculate the length of the goods train by multiplying the relative speed in meters per second by the time it takes to pass the man.

💡First, I need to solve the congruence equation ( x + 6721 equiv 3458 pmod{12} ). To isolate ( x ), I subtract 6721 from both sides, which gives ( x equiv 3458 - 6721 pmod{12} ).Calculating the difference ( 3458 - 6721 ) results in ( -3263 ). Next, I reduce ( -3263 ) modulo 12. Dividing ( -3263 ) by 12 gives a remainder of 9, so ( x equiv 9 pmod{12} ).Now, I need to find the smallest positive integer ( x ) that satisfies ( x equiv 9 pmod{12} ) and is divisible by 5. I express ( x ) as ( 12k + 9 ) where ( k ) is an integer. For ( x ) to be divisible by 5, ( 12k + 9 ) must be congruent to 0 modulo 5.Simplifying, I get ( 2k + 4 equiv 0 pmod{5} ), which leads to ( 2k equiv 1 pmod{5} ). The multiplicative inverse of 2 modulo 5 is 3, so multiplying both sides by 3 gives ( k equiv 3 pmod{5} ). The smallest positive integer ( k ) that satisfies this is 3.Substituting ( k = 3 ) back into the expression for ( x ), I find ( x = 12(3) + 9 = 45 ). Therefore, the smallest positive integer ( x ) that meets both conditions is 45.