To solve the system of equations
[tex]\[
\left\{\begin{array}{l}
y = 2x - 6 \\
4x - 2y = 14
\end{array}\right.
\][/tex],
we'll use the substitution method. The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation.
First, let's consider the first equation:
[tex]\[ y = 2x - 6 \][/tex]
We can substitute [tex]\( y \)[/tex] from the first equation into the second equation.
Second equation:
[tex]\[ 4x - 2y = 14 \][/tex]
Substituting [tex]\( y = 2x - 6 \)[/tex] into the second equation:
[tex]\[ 4x - 2(2x - 6) = 14 \][/tex]
Now, distribute the [tex]\(-2\)[/tex]:
[tex]\[ 4x - 4x + 12 = 14 \][/tex]
Simplify the equation:
[tex]\[ 0 + 12 = 14 \][/tex]
[tex]\[ 12 = 14 \][/tex]
This statement, [tex]\( 12 = 14 \)[/tex], is clearly false. A false statement like this indicates that there are no values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that can satisfy both equations simultaneously.
Therefore, the system of equations has no solution.
So the correct answer is:
- There is no solution.
