To determine how many solutions exist for the given system of linear equations:
[tex]$
\left\{\begin{array}{c}
4x - 5y = 5 \\
-0.08x + 0.10y = 0.10
\end{array}\right.
$[/tex]
We will analyze the system step-by-step.
### Step 1: Understand the Equations
We have two linear equations involving two variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
1. [tex]\( 4x - 5y = 5 \)[/tex]
2. [tex]\( -0.08x + 0.10y = 0.10 \)[/tex]
### Step 2: Simplify Equations (if needed)
First, simplify the second equation by eliminating the decimals. Multiply the entire equation by 1000 to make it easier to work with:
[tex]\[ -0.08x + 0.10y = 0.10 \][/tex]
[tex]\[ -0.08 \cdot 1000 x + 0.10 \cdot 1000 y = 0.10 \cdot 1000 \][/tex]
[tex]\[ -80x + 100y = 100 \][/tex]
Thus, our system can be written as:
1. [tex]\( 4x - 5y = 5\)[/tex]
2. [tex]\( -80x + 100y = 100 \)[/tex]
### Step 3: Convert to Standard Form if Needed
We see that both equations are already in standard form [tex]\( Ax + By = C \)[/tex].
### Step 4: Solve the System
To solve such a system, we could use methods like substitution or elimination, or analyze the coefficients to identify special cases like parallel lines (no solutions) or overlapping lines (infinite solutions).
### Step 5: Analyze for Consistency
We'll transform one of the equations to match the form of the other equation.
The second equation:
[tex]\[ -80x + 100y = 100 \][/tex]
Divide the second equation by 20 to simplify:
[tex]\[ -4x + 5y = 5 \][/tex]
Now the system is:
1. [tex]\( 4x - 5y = 5 \)[/tex]
2. [tex]\( -4x + 5y = 5 \)[/tex]
Let's add these two simplified equations:
[tex]\[ (4x - 5y) + (-4x + 5y) = 5 + 5 \][/tex]
[tex]\[ 0 = 10 \][/tex]
### Conclusion
The statement [tex]\( 0 = 10 \)[/tex] is a contradiction, indicating that the system of equations is inconsistent. Therefore, the two lines are parallel and do not intersect at any point.
Thus, we conclude that the system of equations has no solutions.
Answer:
The system of equations has no solutions.
