Solve the system by the method of your choice.
left brace Start 2 By 1 Matrix 1st Row 1st Column 0.3 x plus 0.4 y equals negative 0.5 2nd Row 1st Column 0.5 x minus 0.3 y equals negative 1.8 EndMatrix
 
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
There is exactly one solution. The solution set is StartSet nothing EndSet
  
. 
​(Simplify your answer. Type an ordered​ pair.)
B.
There are infinitely many solutions. The solution set is StartSet left parenthesis x comma y right parenthesis | 0.3 x plus 0.4 y equals negative 0.5 EndSet or StartSet left parenthesis x comma y right parenthesis | 0.5 x minus 0.3 y equals negative 1.8 EndSet.
C.
The solution set is empty set.

To solve this system of equations, we will use the method of substitution or elimination. Let’s use the elimination method in this case. The system of equations is:

\[
\begin{aligned}
0.3x + 0.4y &= -0.5 \quad \text{(1)} \\
0.5x - 0.3y &= -1.8 \quad \text{(2)}
\end{aligned}
\]

### Step 1: Eliminate one variable
We can eliminate one of the variables by multiplying the equations by suitable factors to align the coefficients.

- Multiply equation (1) by \(5\) to eliminate decimals:
\[
5(0.3x + 0.4y) = 5(-0.5) 
\]
\[
1.5x + 2y = -2.5 \quad \text{(3)}
\]

- Multiply equation (2) by \(3\):
\[
3(0.5x - 0.3y) = 3(-1.8)
\]
\[
1.5x - 0.9y = -5.4 \quad \text{(4)}
\]

### Step 2: Subtract the equations
Now subtract equation (4) from equation (3) to eliminate \(x\):
\[
(1.5x + 2y) - (1.5x - 0.9y) = (-2.5) - (-5.4)
\]
\[
1.5x - 1.5x + 2y + 0.9y = -2.5 + 5.4
\]
\[
2.9y = 2.9
\]

### Step 3: Solve for \(y\)
\[
y = \frac{2.9}{2.9} = 1
\]

### Step 4: Substitute \(y = 1\) into one of the original equations
Substitute \(y = 1\) into equation (1):
\[
0.3x + 0.4(1) = -0.5
\]
\[
0.3x + 0.4 = -0.5
\]
\[
0.3x = -0.5 - 0.4
\]
\[
0.3x = -0.9
\]
\[
x = \frac{-0.9}{0.3} = -3
\]

### Conclusion:
The solution to the system is \(x = -3\) and \(y = 1\). Therefore, the solution set is:

\[
\boxed{(-3, 1)}
\]

This corresponds to option **A**: There is exactly one solution, \((-3, 1)\).

---

Would you like more details on any step or have any questions? 

Here are 5 related questions to explore:

1. Can the substitution method be used to solve this system? How would it work?
2. How do you determine when a system has infinitely many solutions?
3. What are the conditions for a system of linear equations to have no solution?
4. Can this system be solved graphically? How would the solution appear on a graph?
5. How does the elimination method compare with Gaussian elimination for this system?

**Tip:** Always simplify equations by eliminating decimals or fractions early on for easier calculations.

Solve the system by the method of your choice.
Start 2x1 Matrix:
1st Row: 0.3x + 0.4y = -0.5
2nd Row: 0.5x - 0.3y = -1.8

Part 1
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. There is exactly one solution. The solution set is ___. (Simplify your answer. Type an ordered pair.)
B. There are infinitely many solutions. The solution set is StartSet left parenthesis x comma y right parenthesis | 0.3 x plus 0.4 y equals negative 0.5 EndSet or StartSet left parenthesis x comma y right parenthesis | 0.5 x minus 0.3 y equals negative 1.8 EndSet.
C. The solution set is the empty set.

Learn how to solve a system of linear equations involving decimals using the elimination method. This problem helps students practice eliminating variables to find the solution to equations 0.3x + 0.4y = -0.5 and 0.5x - 0.3y = -1.8. Suitable for high school students in Grades 9-12.

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