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Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. (If the system has no solution, enter NONE for each answer. If the system is dependent, enter INFINITELY MANY for each answer.){(*x*, *y*)} = {(  ,  )}

We are given the following system of equations:

\[
\begin{cases}
x + y = 1000 \\
0.17x + 0.18y = 176
\end{cases}
\]

We will use the substitution method to solve this system.

### Step 1: Solve the first equation for one variable
From the first equation, solve for \( y \):

\[
y = 1000 - x
\]

### Step 2: Substitute into the second equation
Substitute \( y = 1000 - x \) into the second equation \( 0.17x + 0.18y = 176 \):

\[
0.17x + 0.18(1000 - x) = 176
\]

Now, simplify and solve for \( x \):

\[
0.17x + 0.18(1000) - 0.18x = 176
\]

\[
0.17x + 180 - 0.18x = 176
\]

\[
-0.01x + 180 = 176
\]

\[
-0.01x = 176 - 180
\]

\[
-0.01x = -4
\]

\[
x = \frac{-4}{-0.01} = 400
\]

### Step 3: Solve for \( y \)
Now that we know \( x = 400 \), substitute this back into the equation for \( y \):

\[
y = 1000 - 400 = 600
\]

### Final Solution
The solution to the system is:

\[
(x, y) = (400, 600)
\]

Would you like further details on this solution, or do you have any questions?

### Related Questions:
1. What would happen if we had different coefficients in the second equation?
2. Can this system be solved graphically? If so, how?
3. What is the geometric interpretation of the solution?
4. How would the solution change if we had the equation \( 0.17x + 0.18y = 200 \)?
5. Can we use the elimination method on this system? If yes, how?

**Tip:** Always try to simplify equations before substituting or eliminating to avoid errors in large expressions.

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. (If the system has no solution, enter NONE for each answer. If the system is dependent, enter INFINITELY MANY for each answer.)

Learn how to solve a system of linear equations using the substitution method. This problem involves solving the system x + y = 1000 and 0.17x + 0.18y = 176, and is suitable for students in Grades 9-12.

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