To estimate the solution to the given system of equations:
[tex]\[
\left\{\begin{array}{l}
7 x - y = 7 \\
x + 2 y = 6
\end{array}\right.
\][/tex]
Let's solve the equations step by step.
### Step 1: Solve one of the equations for one variable
Let's solve the second equation for [tex]\( x \)[/tex]:
[tex]\[ x + 2y = 6 \][/tex]
[tex]\[ x = 6 - 2y \][/tex]
### Step 2: Substitute the expression into the other equation
Now, substitute [tex]\( x = 6 - 2y \)[/tex] into the first equation:
[tex]\[ 7(6 - 2y) - y = 7 \][/tex]
[tex]\[ 42 - 14y - y = 7 \][/tex]
[tex]\[ 42 - 15y = 7 \][/tex]
### Step 3: Solve for [tex]\( y \)[/tex]
[tex]\[ 42 - 15y = 7 \][/tex]
[tex]\[ -15y = 7 - 42 \][/tex]
[tex]\[ -15y = -35 \][/tex]
[tex]\[ y = \frac{-35}{-15} \][/tex]
[tex]\[ y = \frac{35}{15} \][/tex]
[tex]\[ y = \frac{7}{3} \][/tex]
[tex]\[ y = 2 \frac{1}{3} \][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
Substitute [tex]\( y = 2 \frac{1}{3} \)[/tex] back into the expression for [tex]\( x \)[/tex]:
[tex]\[ x = 6 - 2 \left(2 \frac{1}{3}\right) \][/tex]
[tex]\[ x = 6 - 2 \cdot \frac{7}{3} \][/tex]
[tex]\[ x = 6 - \frac{14}{3} \][/tex]
[tex]\[ x = 6 - 4 \frac{2}{3} \][/tex]
[tex]\[ x = 1 \frac{1}{3} \][/tex]
### Final Answer
Therefore, the solution to the system of equations is:
[tex]\[
x = 1 \frac{1}{3}, \quad y = 2 \frac{1}{3}
\][/tex]
So, the correct answer is:
(C) [tex]\( x = 1 \frac{1}{3}, y = 2 \frac{1}{3} \)[/tex]
