How do you solve the simultaneous equations \[x - 7y = 9,7y = 3x + 1\]

Hint: We use the substitution method to solve two linear equations given in the question. We find the value of x from the first equation in terms of y and substitute in the second equation which becomes an equation in y entirely. Solve for the value of y and substitute back the value of y to obtain the value of x.

Complete step by step solution:
We have two linear equations \[x - 7y = 9\] and \[7y = 3x + 1\]
Let us solve the second equation to obtain the value of y in terms of x.
We have \[x - 7y = 9\]
We can shift the values except x to right hand side of the equation
\[ \Rightarrow x = 9 + 7y\] … (1)
Now we substitute the value of \[x = 9 + 7y\]from equation (1) in the first linear equation.
Substitute \[x = 9 + 7y\]in \[7y = 3x + 1\]
\[ \Rightarrow 7y = 3(9 + 7y) + 1\]
Multiply terms outside the bracket to the terms inside the bracket
\[ \Rightarrow 7y = 27 + 21y + 1\]
Add constant values together on right hand side of the equation
\[ \Rightarrow 7y = 28 + 21y\]
Bring all variables to one side and all constants to other side of the equation
\[ \Rightarrow 7y - 21y = 28\]
\[ \Rightarrow - 14y = 28\]
Cancel same factors from both sides of the equation
\[ \Rightarrow y = - 2\]
Put the value of ‘y’ in equation (1) to calculate ‘x’
\[ \Rightarrow x = 9 + 7( - 2)\]
\[ \Rightarrow x = 9 - 14\]
\[ \Rightarrow x = - 5\]
So, the values of \[x = - 5\] and \[y = - 2\]
\[\therefore \] System of linear equations \[x - 7y = 9,7y = 3x + 1\] has solution \[x = - 5\] and \[y = - 2\]

Note: Many students forget to change the sign of the value when shifting from one side of the equation to another, keep in mind the sign always becomes opposite when the value is shifted to the opposite side of the equation.
Alternate method:
We can use elimination methods: Subtract one equation from another equation directly. Subtract corresponding values of one equation from another.
We have equations \[x - 7y = 9,7y = 3x + 1\]
We can write the set of equations as:
\[x - 7y = 9,3x - 7y = - 1\]
Now subtract one equation from another:
\[
  x - 7y = 9 \\
  \underline {3x - 7y = - 1} \\
   - 2x = 10 \\
 \]
i.e. \[x = - 5\]
Now substitute the value of ‘x’ obtained in any of the linear equations and calculate the value of ‘y’’.
\[ \Rightarrow ( - 5) - 7y = 9\]
Shift constant values to right hand side of the equation
\[ \Rightarrow - 7y = 9 + 5\]
Add the value on right hand side of the equation
\[ \Rightarrow - 7y = 14\]
Cancel same factors from both sides of the equation
\[ \Rightarrow y = - 2\]
So, the values of \[x = - 5\] and \[y = - 2\]
\[\therefore \] System of linear equations \[x - 7y = 9,7y = 3x + 1\] has solution \[x = - 5\] and \[y = - 2\]