X + 0.99999y = 2.99999
0.99999x + y = 2.99998


We are given the system of linear equations:

\[
1) \ X + 0.99999y = 2.99999
\]
\[
2) \ 0.99999x + y = 2.99998
\]

We can solve this system using either substitution or elimination. Let’s use substitution:

### Step 1: Solve the first equation for \( X \):

From equation (1):
\[
X = 2.99999 - 0.99999y
\]

### Step 2: Substitute into the second equation:

Now substitute \( X = 2.99999 - 0.99999y \) into the second equation:

\[
0.99999(2.99999 - 0.99999y) + y = 2.99998
\]

### Step 3: Simplify the equation:

First, distribute \( 0.99999 \) on the left-hand side:

\[
0.99999 \times 2.99999 - 0.99999^2 \times y + y = 2.99998
\]

Calculate \( 0.99999 \times 2.99999 \) and \( 0.99999^2 \):

\[
2.99997 - 0.99998y + y = 2.99998
\]

### Step 4: Combine like terms:

Combine the \( y \)-terms:

\[
2.99997 + 0.00001y = 2.99998
\]

### Step 5: Solve for \( y \):

Subtract \( 2.99997 \) from both sides:

\[
0.00001y = 0.00001
\]

Divide both sides by \( 0.00001 \):

\[
y = 1
\]

### Step 6: Solve for \( X \):

Now substitute \( y = 1 \) back into the expression for \( X \):

\[
X = 2.99999 - 0.99999(1)
\]
\[
X = 2.99999 - 0.99999
\]
\[
X = 2
\]

### Final Solution:
The solution to the system is:
\[
X = 2, \ y = 1
\]

Would you like more details on any step? 

Here are 5 related questions:
1. How do we solve systems of equations using elimination instead of substitution?
2. What are other methods to solve systems of linear equations?
3. How does numerical precision affect the solution of a system of equations?
4. Can we solve this system graphically and how would that look?
5. What is the significance of the coefficients being close to 1?

**Tip:** When working with numbers that are very close to 1, like 0.99999, pay close attention to precision, as rounding errors can affect results.

X + 0.99999y = 2.99999, 0.99999x + y = 2.99998

Learn how to solve the system of linear equations X + 0.99999y = 2.99999 and 0.99999x + y = 2.99998 using the substitution method. This problem involves key algebraic concepts, including solving systems of equations and applying the distributive property. Suitable for high school students in Grades 9-12.

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