Determine if the point [tex]\((5, -6)\)[/tex] is a solution to the system of equations:
[tex]\[ \begin{array}{l}
6x + 3y = 12 \\
4x + y = 14
\end{array} \][/tex]



Answer :

Let's solve the system of linear equations:

[tex]\[ \begin{cases} 6x + 3y = 12 \\ 4x + y = 14 \end{cases} \][/tex]

### Step 1: Express one variable in terms of the other
First, we can solve the second equation for [tex]\( y \)[/tex]:

[tex]\[ 4x + y = 14 \\ y = 14 - 4x \][/tex]

### Step 2: Substitute [tex]\( y \)[/tex] in the other equation
Next, substitute this expression for [tex]\( y \)[/tex] in the first equation:

[tex]\[ 6x + 3(14 - 4x) = 12 \][/tex]

### Step 3: Simplify and solve for [tex]\( x \)[/tex]
Simplify the equation:

[tex]\[ 6x + 42 - 12x = 12 \\ -6x + 42 = 12 \\ -6x = 12 - 42 \\ -6x = -30 \\ x = 5 \][/tex]

### Step 4: Solve for [tex]\( y \)[/tex]
Now that we have [tex]\( x = 5 \)[/tex], substitute this value back into the equation [tex]\( y = 14 - 4x \)[/tex]:

[tex]\[ y = 14 - 4(5) \\ y = 14 - 20 \\ y = -6 \][/tex]

### Conclusion
The solution to the system of equations is:

[tex]\[ x = 5, \, y = -6 \][/tex]

Thus, the coordinates of the solution are [tex]\( \boxed{(5, -6)} \)[/tex].