To solve the system of linear equations
[tex]\[ \begin{cases}
x + 2y - 3 = 0 \\
5x + 10y + 1 = 0
\end{cases} \][/tex]
we need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations simultaneously. Let's proceed step by step:
1. Simplify the Equations:
Start by isolating [tex]\( x \)[/tex] in the first equation:
[tex]\[ x + 2y - 3 = 0 \implies x = 3 - 2y \][/tex]
2. Substitute [tex]\( x \)[/tex] into the Second Equation:
Substitute [tex]\( x = 3 - 2y \)[/tex] into the second equation:
[tex]\[ 5(3 - 2y) + 10y + 1 = 0 \][/tex]
3. Simplify the Second Equation:
Simplifying [tex]\( 5(3 - 2y) + 10y + 1 \)[/tex]:
[tex]\[ 15 - 10y + 10y + 1 = 0 \][/tex]
[tex]\[ 15 + 1 = 0 \][/tex]
[tex]\[ 16 = 0 \][/tex]
4. Analyze the Result:
The simplification leads to the equation [tex]\( 16 = 0 \)[/tex], which is a contradiction. We obtained a false statement indicating that the system of equations has no solution.
Hence, the system of equations:
[tex]\[ \begin{cases}
x + 2y - 3 = 0 \\
5x + 10y + 1 = 0
\end{cases} \][/tex]
has no solution. The equations are inconsistent.
