Let's define the two numbers as follows:

Let:

• $x$ = one number
• $y$ = the other number (the lesser one)

From the problem, we know two things:

1. The sum of the two numbers is 17: x + y = 17 \tag{1}

2. One number is 3 less than $\frac{2}{3}$ of the other number: x = \frac{2}{3}y - 3 \tag{2}

Solving the system of equations:

Step 1: From equation (2), substitute $x = \frac{2}{3}y - 3$ into equation (1):

$\left( \frac{2}{3}y - 3 \right) + y = 17$

Step 2: Combine like terms: $\frac{2}{3}y + y - 3 = 17$

Step 3: Express $y$ with a common denominator: $\frac{2}{3}y + \frac{3}{3}y - 3 = 17$ $\frac{5}{3}y - 3 = 17$

Step 4: Isolate $y$: $\frac{5}{3}y = 17 + 3$ $\frac{5}{3}y = 20$

Step 5: Solve for $y$: $y = 20 \times \frac{3}{5}$ $y = 12$

So, the lesser number is $\boxed{12}$.

Verifying the solution:

If $y = 12$, then from equation (1): $x + 12 = 17 \quad \Rightarrow \quad x = 5$ And from equation (2), we check: $x = \frac{2}{3}(12) - 3 = 8 - 3 = 5$ Both conditions hold true, so the solution is correct.

Would you like more details on this solution or have any questions?

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Here are some related questions:

1. How would the solution change if the sum of the two numbers was 20?
2. What if one number was 4 less than half the other?
3. Can you set up a similar equation with different coefficients?
4. How do you graph the system of equations used in this problem?
5. How can we solve this using substitution or elimination?

Tip: Always check your solution by substituting it back into the original equations to avoid mistakes.