You have a system of two linear equations:

1. \( y = -2x + 9 \)
2. \( y = \frac{5}{2} - 9 \)

To find the solution, let's solve them by equating the two expressions for \( y \).

### Step 1: Simplify the second equation
The second equation is:
\[
y = \frac{5}{2} - 9
\]
First, convert \( -9 \) to a fraction with a denominator of 2:
\[
-9 = -\frac{18}{2}
\]
Now, subtract:
\[
y = \frac{5}{2} - \frac{18}{2} = \frac{5 - 18}{2} = \frac{-13}{2}
\]
So the second equation simplifies to:
\[
y = -\frac{13}{2}
\]

### Step 2: Set the equations equal
Since both expressions are equal to \( y \), set the right-hand sides equal to each other:
\[
-2x + 9 = -\frac{13}{2}
\]

### Step 3: Solve for \( x \)
To solve for \( x \), first subtract 9 from both sides:
\[
-2x = -\frac{13}{2} - 9
\]
Convert 9 to a fraction with a denominator of 2:
\[
9 = \frac{18}{2}
\]
Now, combine the fractions:
\[
-2x = -\frac{13}{2} - \frac{18}{2} = -\frac{31}{2}
\]
Divide both sides by -2 to solve for \( x \):
\[
x = \frac{-\frac{31}{2}}{-2} = \frac{31}{4}
\]

### Step 4: Find \( y \)
Plug the value of \( x \) back into one of the original equations. Using \( y = -2x + 9 \):
\[
y = -2\left(\frac{31}{4}\right) + 9 = -\frac{62}{4} + 9 = -\frac{62}{4} + \frac{36}{4} = \frac{-62 + 36}{4} = \frac{-26}{4} = -\frac{13}{2}
\]
So, \( y = -\frac{13}{2} \).

### Final Solution:
The solution to the system of equations is:
\[
x = \frac{31}{4}, \quad y = -\frac{13}{2}
\]

Would you like further details or have any questions?

Here are 8 related questions to explore:
1. How do you solve a system of linear equations by substitution?
2. What is the graphical interpretation of the solution to this system of equations?
3. How can you verify the solution to a system of equations?
4. What are the different methods to solve a system of linear equations?
5. How do you solve a system of equations using the elimination method?
6. What happens when a system of equations has no solution?
7. What does it mean for a system of equations to be dependent?
8. How do you solve a system of equations using matrices?

**Tip:** Always check your final answer by substituting it back into the original equations to ensure accuracy.

Learn how to solve a system of linear equations -2x + 9 = 5/2 - 9 with detailed step-by-step solutions. This problem involves understanding linear equations and fraction arithmetic. Suitable for high school students.

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