Question

How do you solve the following linear system: $4x + 3y = 8$, $x - 2y = 13$?

Answer 1

Hint: This problem deals with solving the values of the variables $x$ and $y$, given there is one pair of linear equations. Here the solutions of the given pair of linear equations should be obtained by the help of the substitution method. This substitution method includes substituting one variable from one equation into another equation.

Complete step-by-step solution:
Given two pairs of linear equations which are given by $4x + 3y = 8$ and $x - 2y = 13$.
Now let us consider the equation $4x + 3y = 8$ as the first equation whereas the equation $x - 2y = 13$ is the second equation.
Now consider the second equation which is $x - 2y = 13$, as shown below:
\[ \Rightarrow x - 2y = 13\]
Extracting the expression for $x$, by transferring the term $2y$ to the right hand side of the above equation as shown below:
\[ \Rightarrow x = 13 + 2y\]
Now substituting the expression of $x$ in the first equation which is $4x + 3y = 8$.
Consider the first equation as shown below:
$ \Rightarrow 4x + 3y = 8$
Substituting the expression \[x = 13 + 2y\] in the above equation as shown below:
$ \Rightarrow 4\left( {13 + 2y} \right) + 3y = 8$
$ \Rightarrow 52 + 8y + 3y = 8$
Simplifying the above expression as shown below:
$ \Rightarrow 52 + 11y = 8$
Rearranging the $y$ terms to one side and the constants to the other side as shown below:
$ \Rightarrow 11y = - 44$
$\therefore y = - 4$
Now finding the value of $x$, from the expression \[x = 13 + 2y\] as shown below:
$ \Rightarrow x = 13 + 2\left( { - 4} \right)$
\[\therefore x = 5\]
The solutions of the equation $4x + 3y = 8$ and $x - 2y = 13$ are $x = 5$ and $y = - 4$.

Note: Please note that the solutions of the above pair of linear equations can also be obtained by the method of elimination. That is by multiplying one of the equations with a number and cancelling out the term, finding the value of one variable and from it we find the value of another variable as well, and hence finding the value of the variables $x$ and $y$ respectively.