How do you solve the simultaneous equations $5x + 2y = 12$ and $ - 6x - 2y = - 14$ ?

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.

Complete step by step solution:
In the question, we are given a couple of simultaneous linear equations in two variables.
$y = 2x + 3$
$y = 3x + 1$
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. In elimination method, we try to eliminate any one of the two variables by adding or subtracting one equation from another.
So, $5x + 2y = 12$ and $ - 6x - 2y = - 14$.
Adding both the equations, we get
 $5x + 2y - 6x - 2y = 12 - 14$
$ \Rightarrow - x = - 2$
$ \Rightarrow x = 2$
Now putting the value of x obtained into any one of the equations to find the value of y.
Hence, we have, $5x + 2y = 12$
$ \Rightarrow 5\left( 2 \right) + 2y = 12$
$ \Rightarrow 2y + 10 = 12$
Shifting from left side of equation to right side of equation, we get,
$ \Rightarrow 2y = 12 - 10$
$ \Rightarrow 2y = 2$
Dividing both sides of equation by $2$, we get,
$ \Rightarrow y = 1$
So, value of y is $1$
Therefore ,solution of the simultaneous linear equations $5x + 2y = 12$ and $ - 6x - 2y = - 14$ is $x = 2$ and $y = 1$

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.