stuck on this problem. i started with x-2y+4z=23.

here is the problem:
Three men, each having denarii, found a purse containing 23 denarii. The first man said to the second, “If I take this purse, I will have twice as much as you.” The second said to the third, “If I take this purse, I will have three times as much as you.” The third man said to the first, “If I take this purse, I will have four times as much as you.” How many denarii did each man have?

Try this, which is a better rendering of the given statements:

x+23 = 2y
y+23 = 3z
z+23 = 4x

I'LL TRY THIS THANKS

To solve this problem, we need to translate the given information into equations.

Let's assume that the first man has x denarii, the second man has y denarii, and the third man has z denarii.

From the given information, we can create equations based on the statements of each man:

1) "If I take this purse, I will have twice as much as you."
This statement translates to x + 23 = 2(y - 23).

2) "If I take this purse, I will have three times as much as you."
This statement translates to y + 23 = 3(z - 23).

3) "If I take this purse, I will have four times as much as you."
This statement translates to z + 23 = 4(x - 23).

Now, we have three equations based on the given information:

Equation 1: x + 23 = 2(y - 23)
Equation 2: y + 23 = 3(z - 23)
Equation 3: z + 23 = 4(x - 23)

To solve this system of equations, we can use the method of substitution or elimination.

One possible way to solve this system is by substituting variables:

From Equation 1, we can rearrange it to express x in terms of y:
x = 2(y - 23) - 23 = 2y - 69

Substitute this expression for x into Equation 3:
z + 23 = 4((2y - 69) - 23)

Now, simplify:
z + 23 = 4(2y - 92)
z + 23 = 8y - 368

Rearrange this equation to express z in terms of y:
z = 8y - 391

Now, substitute these expressions for x and z into Equation 2:
y + 23 = 3((8y - 391) - 23)

Simplify:
y + 23 = 3(8y - 414)
y + 23 = 24y - 1242

Rearrange this equation to express y in terms of a constant:
23 + 1242 = 24y - y
y = 1265/23

Therefore, the second man has 1265/23 denarii.

Now, substitute this value for y into Equation 1 to find x:
x = 2(1265/23 - 23) - 23 = 1026/23

Hence, the first man has 1026/23 denarii.

Finally, substitute the values of y and x into Equation 3 to find z:
z + 23 = 4((2 * 1265/23) - 23)
z + 23 = 4(2530/23 - 23)
z + 23 = 10120/23 - 92
z + 23 = 10120/23 - 92(23/23)
z + 23 = (10120 - 2116)/23
z + 23 = 8004/23
z = 8004/23 - 23(23/23)
z = (8004 - 529)/23
z = 7475/23

Therefore, the third man has 7475/23 denarii.

To answer the problem, the first man has 1026/23 denarii, the second man has 1265/23 denarii, and the third man has 7475/23 denarii.