To solve the given system of equations:
[tex]\[
\left\{\begin{array}{l}
-x + 3y + 5z = 0 \\
5x - 15y - 25z = 2
\end{array}\right.
\][/tex]
we can proceed as follows:
1. Simplify the second equation:
We observe that the second equation can be simplified by dividing every term by 5:
[tex]\[
5x - 15y - 25z = 2
\][/tex]
Dividing each term by 5:
[tex]\[
x - 3y - 5z = \frac{2}{5}
\][/tex]
2. Rewrite the system with the simplified second equation:
Now, our system looks like this:
[tex]\[
\left\{\begin{array}{l}
-x + 3y + 5z = 0 \\
x - 3y - 5z = \frac{2}{5}
\end{array}\right.
\][/tex]
3. Add the two equations to eliminate [tex]\( x \)[/tex]:
We add the left-hand sides and the right-hand sides of the two equations:
[tex]\[
(-x + 3y + 5z) + (x - 3y - 5z) = 0 + \frac{2}{5}
\][/tex]
Simplifying the left-hand side:
[tex]\[
-x + x + 3y - 3y + 5z - 5z = \frac{2}{5}
\][/tex]
This simplifies to:
[tex]\[
0 = \frac{2}{5}
\][/tex]
4. Analyze the resulting equation:
The equation:
[tex]\[
0 = \frac{2}{5}
\][/tex]
is a contradiction, as 0 cannot equal [tex]\(\frac{2}{5}\)[/tex]. This indicates that there is no set of values for [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] that can satisfy both equations simultaneously.
5. Conclusion:
Based on the contradiction obtained in step 3, we conclude that the system of equations has no solution.
