Determine whether the system of linear equations has one and only one solution, infinitely many solutions, or no solution.
2x	 − 	y	 = 	3
3x	 + 	y	 = 	−5
 
one and only one solution
infinitely many solutions    
no solution

Find the solution using elimination, if one exists. (If there are infinitely many solutions, express x and y in terms of the parameter t. If there is no solution, enter NO SOLUTION.)

Let's solve the system of equations using the elimination method:

The given system is:
1. \( 2x - y = 3 \)  (Equation 1)
2. \( 3x + y = -5 \)  (Equation 2)

### Step 1: Add the two equations to eliminate \( y \).

To do this, add Equation 1 and Equation 2:
\[
(2x - y) + (3x + y) = 3 + (-5)
\]

Simplifying the left and right sides:
\[
5x = -2
\]

### Step 2: Solve for \( x \).

\[
x = \frac{-2}{5}
\]

### Step 3: Substitute \( x = \frac{-2}{5} \) into one of the original equations.

Let's substitute \( x = \frac{-2}{5} \) into Equation 1:
\[
2x - y = 3
\]
\[
2 \times \frac{-2}{5} - y = 3
\]
\[
\frac{-4}{5} - y = 3
\]

### Step 4: Solve for \( y \).

First, add \( \frac{4}{5} \) to both sides:
\[
- y = 3 + \frac{4}{5}
\]
\[
- y = \frac{15}{5} + \frac{4}{5}
\]
\[
- y = \frac{19}{5}
\]

Now, multiply both sides by \( -1 \):
\[
y = \frac{-19}{5}
\]

### Conclusion:

The solution to the system is:
\[
x = \frac{-2}{5}, \quad y = \frac{-19}{5}
\]

This system has **one and only one solution**.

---

Do you want more details or have any questions? Here are some additional questions to consider:
1. How do you interpret the graphical representation of this system?
2. What if we used substitution instead of elimination—would the solution be the same?
3. Can you explain how to determine if a system has infinitely many solutions?
4. What happens if we change one of the equations—how will that affect the solution?
5. Can you apply this method to a 3-variable system of equations?

**Tip**: Always check for consistency in the system of equations by looking at the coefficients; it can hint whether there's no solution or infinitely many solutions.

Determine whether the system of linear equations has one and only one solution, infinitely many solutions, or no solution. 2x − y = 3, 3x + y = −5. Find the solution using elimination, if one exists.

Learn how to solve a system of linear equations 2x - y = 3 and 3x + y = -5 using the elimination method. This step-by-step guide explores key algebra concepts and determines whether the system has one solution, infinitely many solutions, or no solution. Suitable for Grades 8-10.

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