Question

How do you solve the system of equations $x + 2y = - 2$ and $3x + 4y = 6$?

Answer 1

Hint: This problem deals with solving the system of linear equations in $x$ and $y$. This is done by adding or subtracting a multiple of one equation to the other equation, in such a way that either the $x$-terms or the $y$-terms cancel out. Then solve for $x$(or $y$, whichever left) and substitute back to get the other coordinate.

Complete step-by-step solution:
Given a pair of linear equations in $x$ and $y$, which are given by $x + 2y = - 2$ and $3x + 4y = 6$,
Now we have to solve these two equations in order to get the values of $x$ and $y$.
Consider the given two equations and rearrange them to solve accordingly.
Consider the first equation, as given below:
$ \Rightarrow \;x + 2y = - 2$
Now consider the second equation, as given below:
$ \Rightarrow 3x + 4y = 6$
Rearranging the first equation, so as to solve this equation together with the second equation.
Multiply the first equation with 3, so as match the coefficient of $x$, so that we can solve both the equations in a simpler way, as shown below:
$ \Rightarrow 3\left( {x + 2y = - 2} \right)$
$ \Rightarrow 3x - 6y = - 6$
Now subtracting the first and second equation, as shown below:
$ \Rightarrow 3x + 6y = - 6$
$ \Rightarrow 3x + 4y = 6$
$ \Rightarrow 6y - \left( {4y} \right) = - 6 - 6$
On simplification of the above two equations, as given below:
$ \Rightarrow 2y = - 12$
$\therefore y = - 6$
Substituting the value of $y = - 6$, in the first equation $x + 2y = - 2$, as given below:
$ \Rightarrow \;x + 2y = - 2$
$ \Rightarrow \;x + 2\left( { - 6} \right) = - 2$
Simplifying the above equations, as given below:
$ \Rightarrow \;x - 12 = - 2$
Grouping the constants to the other side of the equation, gives:
$ \Rightarrow x = - 2 + 12$
$\therefore x = 10$
Hence the values of $x = 10$ and $y = - 6$.

The values of $x$ and $y$ are 10 and -6 respectively.

Note: Here a system of equations is when we have two or more linear equations working together. Here this problem can be done in another way but with a slight change with the method solved here. Here instead of multiplying the first equation with 3, we can multiply it with 2 and solve both the equations, finally ending up the same solution.