To determine the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the given system of equations:
[tex]\[ \left\{\begin{array}{l} 3x + 2y = 14 \\ x = 4y - 2 \end{array}\right. \][/tex]
we will proceed step-by-step:
1. Substitute the second equation into the first equation:
The second equation gives [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ x = 4y - 2 \][/tex]
Substitute [tex]\( x = 4y - 2 \)[/tex] into the first equation:
[tex]\[ 3(4y - 2) + 2y = 14 \][/tex]
2. Simplify and solve for [tex]\( y \)[/tex]:
Expand and combine like terms:
[tex]\[ 12y - 6 + 2y = 14 \][/tex]
Combine the [tex]\( y \)[/tex] terms:
[tex]\[ 14y - 6 = 14 \][/tex]
Add 6 to both sides:
[tex]\[ 14y = 20 \][/tex]
Divide by 14:
[tex]\[ y = \frac{20}{14} = \frac{10}{7} \][/tex]
3. Substitute [tex]\( y = \frac{10}{7} \)[/tex] back into the second equation to solve for [tex]\( x \)[/tex]:
Using [tex]\( x = 4y - 2 \)[/tex]:
[tex]\[ x = 4\left(\frac{10}{7}\right) - 2 \][/tex]
[tex]\[ x = \frac{40}{7} - 2 \][/tex]
[tex]\[ x = \frac{40}{7} - \frac{14}{7} \][/tex]
[tex]\[ x = \frac{26}{7} \][/tex]
Thus, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the system of equations are:
[tex]\[ \left(\frac{26}{7}, \frac{10}{7}\right) \][/tex]
Therefore, the correct answer is:
[tex]\(\left(\frac{26}{7}, \frac{10}{7}\right)\)[/tex]
