How do you solve the simultaneous equations $ 5m - 3n = 19 $ and $ m - 6n = - 7 $ ?

Hint: In the given question, we need to solve two simultaneous equations in two variables. There are various methods to solve two given equations in two variables like substitution method, cross multiplication method, elimination method, matrix method and many more. The equations given in the question can be solved using any one of the above mentioned methods easily. But we will solve the equations using the substitution method as it is the most basic method to solve such a question.

Complete step by step solution:
In the question, we are given a couple of simultaneous linear equations in two variables.
 $ 5m - 3n = 19 - - - - - - - - \left( 1 \right) $
 $ m - 6n = - 7 - - - - - - - - \left( 2 \right) $
In the substitution method, we substitute the value of one variable from an equation into another equation so as to get an equation in only one variable.
Now we put the value of m obtained from the second equation into the first equation.
So, we have, $ m = 6n - 7 $
Putting this value of m in the first question, we get,
 $ \Rightarrow 5\left( {6n - 7} \right) - 3n = 19 $
Opening the brackets and simplifying, we get,
 $ \Rightarrow 30n - 35 - 3n = 19 $
 $ \Rightarrow 27n = 35 + 19 $
Simplifying further, we get,
 $ \Rightarrow 27n = 54 $
Obtaining the value of n by isolating the variable, we get,3
 $ \Rightarrow n = 2 $
So, the value of n is $ 2 $ .
Hence, we put the value of n in the second equation to get the value of m.
So, we get, $ m - 6\left( 2 \right) = - 7 $
Simplifying the expression, we get,
 $ \Rightarrow m = - 7 + 12 $
 $ \Rightarrow m = 5 $
So, the value of m is $ 5 $ .
Therefore, solution of the simultaneous linear equations $ 5m - 3n = 19 $ and $ m - 6n = - 7 $ is $ n = 2 $ and $ m = 5 $
So, the correct answer is “ $ n = 2 $ and $ m = 5 $ ”.

Note: Linear Equation in two variables: A equation consisting of 2 variables having degree one is known as Linear Equation in two variables. Standard form of Linear Equation in two variables is $ ax + by + c = 0 $ where a, b and c are the real numbers and a, b which are coefficients of x and y respectively are not equal to 0.