answer：Okay, so I have this system of equations:[begin{cases}x - y = 5 x + 2y = 3m - 1end{cases}]And I know that the solution to this system also satisfies another equation: (2x + y = 13). I need to find the value of (m). Hmm, let's see. First, maybe I should solve the system of equations for (x) and (y) in terms of (m), and then plug those into the third equation to find (m). That sounds like a plan.Starting with the first equation: (x - y = 5). Maybe I can solve for one variable in terms of the other. Let's solve for (x). So, adding (y) to both sides gives:(x = y + 5)Okay, so now I can substitute this expression for (x) into the second equation. The second equation is (x + 2y = 3m - 1). Substituting (x = y + 5) into that:((y + 5) + 2y = 3m - 1)Simplify the left side:(y + 5 + 2y = 3m - 1)Combine like terms:(3y + 5 = 3m - 1)Now, let's solve for (y). Subtract 5 from both sides:(3y = 3m - 1 - 5)Simplify the right side:(3y = 3m - 6)Divide both sides by 3:(y = m - 2)Alright, so (y = m - 2). Now, since we have (x = y + 5), substitute (y) back into that:(x = (m - 2) + 5)Simplify:(x = m + 3)So now, we have expressions for both (x) and (y) in terms of (m):(x = m + 3)(y = m - 2)Great, now the next step is to use the third equation that the solution satisfies: (2x + y = 13). Let's substitute our expressions for (x) and (y) into this equation.Substituting (x = m + 3) and (y = m - 2) into (2x + y):(2(m + 3) + (m - 2) = 13)Let me expand this:First, distribute the 2 into (m + 3):(2m + 6 + m - 2 = 13)Now, combine like terms:(2m + m + 6 - 2 = 13)That simplifies to:(3m + 4 = 13)Now, solve for (m). Subtract 4 from both sides:(3m = 13 - 4)Which is:(3m = 9)Divide both sides by 3:(m = 3)So, the value of (m) is 3. Let me just double-check my steps to make sure I didn't make a mistake.Starting with the system:1. (x - y = 5) leads to (x = y + 5).2. Substitute into the second equation: (y + 5 + 2y = 3m - 1) simplifies to (3y + 5 = 3m - 1), then (3y = 3m - 6), so (y = m - 2).3. Then (x = (m - 2) + 5 = m + 3).4. Substitute into (2x + y = 13): (2(m + 3) + (m - 2) = 13) becomes (2m + 6 + m - 2 = 13), which is (3m + 4 = 13), so (3m = 9), hence (m = 3).Yep, that seems correct. I don't see any errors in the algebra. So, I think (m = 3) is the right answer.**Final Answer**The value of (m) is boxed{3}.

answer：First, I need to determine the total time William spent traveling from Missouri to his hometown. He left at 7:00 AM and arrived at 8:00 PM. The duration from 7:00 AM to 8:00 PM is 13 hours.Next, I'll calculate the total time he spent during his stops. He had three stops: 25 minutes, 10 minutes, and another 25 minutes. Adding these together gives 60 minutes, which is equivalent to 1 hour.Finally, to find out how much time he spent on the road, I'll subtract the total stop time from the total travel time. So, 13 hours minus 1 hour equals 12 hours.

answer：To find the magnitude of the vector difference between two unit vectors with an angle of π/3 between them, I can use the formula for the magnitude of the difference of two vectors.First, I'll denote the vectors as (overrightarrow{a}) and (overrightarrow{b}), both of which are unit vectors, so their magnitudes are 1. The angle between them is given as (frac{pi}{3}).The formula for the magnitude of (overrightarrow{a} - overrightarrow{b}) is:[|overrightarrow{a} - overrightarrow{b}| = sqrt{|overrightarrow{a}|^2 + |overrightarrow{b}|^2 - 2|overrightarrow{a}||overrightarrow{b}|costheta}]Since both vectors are unit vectors, (|overrightarrow{a}| = |overrightarrow{b}| = 1). Substituting these values into the formula:[|overrightarrow{a} - overrightarrow{b}| = sqrt{1^2 + 1^2 - 2 times 1 times 1 times cosleft(frac{pi}{3}right)}]Next, I'll calculate the cosine of (frac{pi}{3}):[cosleft(frac{pi}{3}right) = frac{1}{2}]Plugging this back into the equation:[|overrightarrow{a} - overrightarrow{b}| = sqrt{1 + 1 - 2 times frac{1}{2}} = sqrt{2 - 1} = sqrt{1} = 1]Therefore, the magnitude of (overrightarrow{a} - overrightarrow{b}) is 1.

answer：To find the 43rd digit after the decimal point in the decimal representation of 1/13, I first need to determine the decimal expansion of 1/13.I'll perform the division of 1 by 13. Dividing 1 by 13 gives a repeating decimal: 0.076923076923..., where the sequence 076923 repeats indefinitely.Next, I'll identify the length of the repeating cycle. The repeating sequence "076923" has 6 digits.To find the position of the 43rd digit within the repeating cycle, I'll divide 43 by 6. The quotient is 7 with a remainder of 1. This means that after 7 full cycles of 6 digits, there is 1 additional digit.Finally, I'll locate the 1st digit in the repeating sequence "076923," which is 0. Therefore, the 43rd digit after the decimal point in the decimal representation of 1/13 is 0.

answer：To find the average of 1/8 and 1/6, I first need to add the two fractions together. Since 1/8 and 1/6 have different denominators, I'll find a common denominator, which is 24. Converting both fractions, 1/8 becomes 3/24 and 1/6 becomes 4/24.Adding these together gives 3/24 + 4/24 = 7/24.Next, to find the average, I'll divide the sum by 2. Dividing 7/24 by 2 results in 7/48.Therefore, the average of 1/8 and 1/6 is 7/48.

answer：Alright, so I've got this problem here where there are several rectangular prisms, each with edge lengths of 2, 3, and 5. They're all aligned in the same direction to form a cube with an edge length of 90. The question is asking how many small rectangular prisms does one diagonal of the cube intersect. The options are 64, 66, 68, or 70. Hmm, okay.First, I need to visualize this setup. There's a big cube made up of smaller rectangular prisms. Each small prism has dimensions 2x3x5, and they're all aligned the same way. So, the big cube has an edge length of 90, which is a multiple of 2, 3, and 5. Let me check: 90 divided by 2 is 45, 90 divided by 3 is 30, and 90 divided by 5 is 18. So, along each edge of the big cube, there are 45 prisms along the 2-unit edge, 30 along the 3-unit edge, and 18 along the 5-unit edge.Wait, actually, no. Since each small prism has edges of 2, 3, and 5, the number of prisms along each edge of the big cube should be 90 divided by each respective edge length. So, along the x-axis, which is length 2, the number of prisms would be 90 / 2 = 45. Similarly, along y-axis (3 units), it's 90 / 3 = 30. And along the z-axis (5 units), it's 90 / 5 = 18. So, the big cube is made up of 45x30x18 small prisms.Now, the problem is about a space diagonal of the cube intersecting how many small prisms. I remember that for a grid of cubes, the number of cubes a diagonal passes through can be calculated using the formula: a + b + c - gcd(a, b) - gcd(b, c) - gcd(c, a) + gcd(a, b, c), where a, b, c are the number of divisions along each axis.But wait, in this case, the prisms are not cubes; they're rectangular. So, does the same formula apply? I think it does, but I need to make sure. The formula is actually for a grid, regardless of the size of each cell, as long as they are axis-aligned. So, if the grid is divided into m x n x p cells, the number of cells a diagonal passes through is m + n + p - gcd(m, n) - gcd(n, p) - gcd(p, m) + gcd(m, n, p). So, in this case, m is 45, n is 30, p is 18.Let me compute that step by step.First, compute m + n + p: 45 + 30 + 18 = 93.Next, compute the gcds:gcd(45, 30): Let's see, 45 and 30. The prime factors of 45 are 3^2 * 5, and for 30, it's 2 * 3 * 5. So, the gcd is 3 * 5 = 15.gcd(30, 18): 30 is 2 * 3 * 5, and 18 is 2 * 3^2. So, gcd is 2 * 3 = 6.gcd(18, 45): 18 is 2 * 3^2, and 45 is 3^2 * 5. So, gcd is 3^2 = 9.Then, subtract these gcds: 15 + 6 + 9 = 30.So, subtracting that from 93: 93 - 30 = 63.Now, compute gcd(m, n, p): gcd(45, 30, 18). Let's see, the common prime factors among all three. 45 is 3^2 * 5, 30 is 2 * 3 * 5, 18 is 2 * 3^2. So, the common factor is 3. So, gcd is 3.Add that back: 63 + 3 = 66.So, according to this formula, the diagonal passes through 66 small prisms.Wait, but hold on. The formula is for a grid where each cell is a unit cube, but in our case, each cell is a rectangular prism with different edge lengths. Does that affect the formula? Hmm.Wait, no, actually, the formula is for the number of grid cells a line passes through in a grid, regardless of the cell sizes, as long as the grid is axis-aligned. So, even if the cells are rectangular, the formula still applies because it's based on the number of divisions along each axis, not the actual lengths. So, 45, 30, 18 are the number of divisions, so the formula should still hold.So, according to that, the answer is 66, which is option B.But wait, let me think again. Maybe I'm misapplying the formula because the prisms are not cubes. Let me check the formula's derivation.The formula for the number of cubes a space diagonal passes through in a grid is indeed m + n + p - gcd(m, n) - gcd(n, p) - gcd(p, m) + gcd(m, n, p). This formula is derived based on the inclusion-exclusion principle, considering the number of times the diagonal crosses a plane perpendicular to each axis, subtracting overlaps where it crosses two planes at once, and adding back where it crosses all three.But in our case, the grid is made up of rectangular prisms, not unit cubes. However, the formula doesn't depend on the size of the prisms, only on the number of divisions along each axis. So, as long as the prisms are axis-aligned, the formula should still be applicable.Therefore, I think the calculation is correct, and the answer is 66, which is option B.But just to be thorough, let me consider an example with smaller numbers to see if the formula holds.Suppose we have a cube made up of 2x2x2 small cubes. Then, the space diagonal would pass through 2 + 2 + 2 - gcd(2,2) - gcd(2,2) - gcd(2,2) + gcd(2,2,2) = 6 - 2 - 2 - 2 + 2 = 6 - 6 + 2 = 2. Wait, that doesn't make sense because in a 2x2x2 cube, the space diagonal passes through all 2 cubes along the way, right? Hmm, actually, no. Wait, in a 2x2x2 cube, the space diagonal starts at one corner and exits at the opposite corner, passing through two cubes. So, the formula gives 2, which is correct.Wait, but in a 1x1x1 cube, the formula would give 1 + 1 + 1 - 1 -1 -1 +1 = 1, which is correct because the diagonal is entirely within the single cube.Another example: a 3x3x3 cube. The formula gives 3 + 3 + 3 - 3 - 3 - 3 + 3 = 9 - 9 + 3 = 3. But in reality, a space diagonal in a 3x3x3 cube passes through 3 cubes, which is correct.Wait, but in a 2x3x4 grid, how many cubes does the diagonal pass through? Let's compute: 2 + 3 + 4 - gcd(2,3) - gcd(3,4) - gcd(4,2) + gcd(2,3,4). So, 2 + 3 + 4 = 9. gcd(2,3)=1, gcd(3,4)=1, gcd(4,2)=2. So, subtract 1 + 1 + 2 = 4. Then, add back gcd(2,3,4)=1. So, total is 9 - 4 + 1 = 6. Let me visualize this: in a 2x3x4 grid, the diagonal would cross through 6 unit cubes. That seems correct.So, the formula seems to hold even when the grid isn't made up of unit cubes, as long as we're talking about the number of divisions along each axis. So, in our original problem, the number of divisions are 45, 30, 18, so the formula gives 45 + 30 + 18 - 15 - 6 - 9 + 3 = 66. So, 66 is the answer.But wait, just to make sure, let me think about how the diagonal intersects the prisms. Each time the diagonal crosses a plane perpendicular to the x-axis, it enters a new prism along the x-direction. Similarly for y and z. So, the total number of prisms intersected is equal to the number of crossings along each axis plus 1 (for the starting prism). But the formula accounts for overlaps where the diagonal crosses two or three planes at once, which are the gcd terms.So, in essence, the formula is counting the number of times the diagonal crosses a plane in each direction, subtracting the overlaps where it crosses two planes at once, and adding back where it crosses all three. So, yes, that should give the correct number of prisms intersected.Therefore, I think the answer is 66, which is option B.**Final Answer**boxed{B}