To determine the nature of solutions for the given system of equations:
[tex]$
\left\{\begin{array}{l}
-5 x - 4 y = -4 \\
25 x + 20 y = 20
\end{array}\right.
$[/tex]
Let’s analyze this step-by-step:
1. Writing the System of Equations:
The system of equations is:
[tex]\[
-5x - 4y = -4 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
25x + 20y = 20 \quad \text{(Equation 2)}
\][/tex]
2. Formulating Equation 2:
Notice that Equation 2 can be rewritten by factoring out 5:
[tex]\[
25x + 20y = 20
\][/tex]
[tex]\[
5(5x + 4y) = 20
\][/tex]
[tex]\[
5x + 4y = 4 \quad \text{(Equation 3 derived from Equation 2)}
\][/tex]
3. Comparing Equation 1 and Equation 3:
The Equation 1 is:
[tex]\[
-5x - 4y = -4
\][/tex]
Comparing Equation 1 and Equation 3 (but rearranged):
[tex]\[
5x + 4y = 4
\][/tex]
Notice that if we multiply Equation 1 by -1, we get:
[tex]\[
-1(-5x - 4y) = -1(-4)
\][/tex]
[tex]\[
5x + 4y = 4
\][/tex]
Which is exactly Equation 3.
4. Conclusion about the System:
Both equations represent the same linear relationship. That is, the second equation is just a scalar multiple of the first equation, meaning they are dependent.
Since the two equations are essentially the same line, there are infinitely many solutions along the line representing this equation. Therefore, every point on the line [tex]\(5x + 4y = 4\)[/tex] is a solution to the system of equations.
Thus, the correct statement that describes this system of equations is:
The system has infinitely many solutions.
