the sum of 2 numbers is 17. one number is 3 less than 2/3 of the other number. what is the lesser number

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.

The sum of 2 numbers is 17. One number is 3 less than 2/3 of the other number. What is the lesser number?

This problem involves solving a system of linear equations to find two numbers that sum to 17, with one number being 3 less than 2/3 of the other. Learn step-by-step how to solve for the lesser number. Suitable for Grades 7-9 students.

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