Okay, let's see. I need to find the modulus of w, where z is given as 1 + i and w is (3z + 1)/(5z + 7). Hmm. Alright, modulus of a complex number is like its distance from the origin in the complex plane, right? So if w is a complex number, |w| is the square root of (a² + b²) where w = a + bi. But calculating that might involve some steps. Let me think about how to approach this.

First, maybe I should substitute z = 1 + i into the expression for w. That makes sense. So let's do that step by step. Let's compute the numerator and the denominator separately. 

Starting with the numerator: 3z + 1. Since z is 1 + i, multiplying by 3 gives 3*(1) + 3*(i) = 3 + 3i. Then adding 1 to that, we get 3 + 1 + 3i = 4 + 3i. So numerator is 4 + 3i. 

Now the denominator: 5z + 7. Again, z is 1 + i. So 5*z is 5*1 + 5*i = 5 + 5i. Adding 7 gives 5 + 7 + 5i = 12 + 5i. So denominator is 12 + 5i.

Therefore, w = (4 + 3i)/(12 + 5i). To find the modulus of w, I can use the property that |w| = |numerator| / |denominator|. Because modulus is multiplicative, right? So |w| = |4 + 3i| / |12 + 5i|. That should simplify things.

Let me compute |4 + 3i| first. The modulus is sqrt(4² + 3²) = sqrt(16 + 9) = sqrt(25) = 5. 

Then compute |12 + 5i|. That's sqrt(12² + 5²) = sqrt(144 + 25) = sqrt(169) = 13. 

Therefore, |w| = 5 / 13. 

Wait, that seems straightforward, but let me verify to make sure I didn't skip any steps or make a mistake. Let me double-check the computations.

First, numerator: 3z + 1. z = 1 + i. 3*(1 + i) = 3 + 3i. Adding 1 gives 4 + 3i. Correct. 

Denominator: 5z + 7. 5*(1 + i) = 5 + 5i. Adding 7 gives 12 + 5i. Correct. 

Then modulus of numerator: sqrt(4² + 3²) = 5. Denominator modulus: sqrt(12² + 5²) = 13. So 5/13. Hmm, seems right.

Alternatively, maybe I can compute w by rationalizing the denominator? Let me try that method as a check. 

So w = (4 + 3i)/(12 + 5i). To rationalize, multiply numerator and denominator by the conjugate of the denominator, which is 12 - 5i. 

So numerator becomes (4 + 3i)(12 - 5i). Let's compute that:

First, 4*12 = 48

4*(-5i) = -20i

3i*12 = 36i

3i*(-5i) = -15i². Since i² = -1, this becomes +15.

So adding all terms: 48 - 20i + 36i +15 = (48 +15) + (-20i +36i) = 63 + 16i.

Denominator: (12 +5i)(12 -5i) = 12² - (5i)² = 144 -25i² = 144 -25*(-1) = 144 +25 = 169.

Therefore, w = (63 +16i)/169. So in terms of a + bi, that's 63/169 + (16/169)i. 

Then |w| is sqrt( (63/169)^2 + (16/169)^2 ) = sqrt( (63² +16²)/169² ) = sqrt( (3969 + 256)/28561 ) = sqrt(4225/28561 ) = sqrt(4225)/sqrt(28561). 

sqrt(4225) is 65, since 65² = 4225. sqrt(28561) is 169, since 169² = 28561. So sqrt(4225)/sqrt(28561) = 65/169. Simplify that fraction: 65 and 169 are both divisible by 13. 65 ÷13=5, 169 ÷13=13. So 5/13. 

Same result! So that confirms it. Therefore, |w| = 5/13. 

Wait, so both methods give the same answer. That gives me more confidence that it's correct. 

Alternatively, since modulus is multiplicative, |w| = |3z +1| / |5z +7|. Which is the same as |4 +3i| / |12 +5i|, which we already computed as 5/13. 

But just to make sure, maybe I should check if there's any mistake in the substitution step. Let me verify again:

z =1 +i. Then 3z = 3*(1 +i) =3 +3i. 3z +1 =3 +3i +1=4 +3i. Correct. 

5z =5*(1 +i)=5 +5i. 5z +7=5 +5i +7=12 +5i. Correct. 

So substitution is correct. Then modulus of numerator is 5, denominator 13. 

Yes, I think that's solid. 

Alternatively, maybe I can use the formula for modulus of a complex number (a + bi) is sqrt(a² + b²). So for w = (4 +3i)/(12 +5i), modulus is |4 +3i| / |12 +5i| =5/13. 

Therefore, the answer should be 5/13. 

I don't see any mistakes here. Both methods of computing w and then its modulus, and directly computing the modulus of numerator and denominator separately give the same result. 

So I think that's the correct answer. 

**Final Answer**
The modulus of \( w \) is \boxed{\dfrac{5}{13}}.
Therefore, the modulus of $w$ is $\boxed{\frac{5}{13}}$.
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