Question

How do you solve this system of equations: $7x + 2y = - 19$ and $ - x + 2y = 21$?

Answer 1

Hint: Here we will proceed by taking one of the equations from the pair and convert it into a third equation. Then we will substitute the newly formed equation into another equation from the given pair of equations, it will give the value of one variable. After that substitute the value in the third equation to get the value of the second variable. Thus, we will get the required values of the equation.

Complete step by step answer:
Linear pairs of equations are equations that can be expressed as \[ax + by + c = 0\] where a, b and c are real numbers and both a, b are non-zero.
In this question, two equations are-
$ \Rightarrow 7x + 2y = - 19$ ….. (1)
$ \Rightarrow - x + 2y = 21$ ….. (2)
Firstly, we will take equation (2) and convert it,
$ \Rightarrow - x + 2y = 21$
Subtract $2y$ from both sides of the equation,
$ \Rightarrow - x + 2y - 2y = 21 - 2y$
Simplify the term,
$ \Rightarrow - x = 21 - 2y$
Multiply both sides by -1,
$ \Rightarrow x = 2y - 21$ ….. (3)
Now we will put the value of x from equation (3) in equation (1),
$ \Rightarrow 7\left( {2y - 21} \right) + 2y = - 19$
Open the bracket and multiply the terms,
$ \Rightarrow 14y - 147 + 2y = - 19$
Move constant part on the right side and simplify the equation,
$ \Rightarrow 16y = 128$
Divide both sides by 16,
$ \Rightarrow y = 8$
Here we will substitute the value of y in equation (3) to get the value of x,
$ \Rightarrow x = 2 \times 8 - 21$
Multiply the terms,
$ \Rightarrow x = 16 - 21$
Simplify the terms on the right side,
$\therefore x = - 5$
Hence, the values of x and y are -5 and 8 respectively.

Note: Whenever we face such types of problems the key concept is to use various methods of variable evaluation either by elimination or by substitution method. These methods will help in getting the right track to evaluate these equations involving two variables and reach the right solution.