Question

How do you solve the system of equations below and give your answer as an ordered pair:
$4x+7y=47$ and $5x-4y=-5$?

Answer 1

We have a system of two linear equations. We will use the method of substitution for solving the given system of linear equations. We will use one equation to write one variable in terms of the other. Then we will substitute this value in the other equation and obtain the value of one variable. We will use this value to calculate the value of the other variable from one of the two equations.

Complete step by step answer:
The given system of linear equations is
$4x+7y=47....(i)$
$5x-4y=-5....(ii)$
Now, we will rearrange equation $(ii)$ to obtain the value of variable $x$ in terms of variable $y$. We will do this in the following manner,
$\begin{aligned} & 5x=4y-5 \\\ & \therefore x=\dfrac{4y-5}{5} \\\ \end{aligned}$
Substituting this value of variable $x$ in equation $(i)$, we get the following,
$4\left( \dfrac{4y-5}{5} \right)+7y=47$
Multiplying the above equation by 5, we get
$4\left( 4y-5 \right)+35y=235$
Simplifying the above equation and solving it for variable $y$, we get the following,
$\begin{aligned} & 16y-20+35y=235 \\\ & \Rightarrow 51y=255 \\\ & \Rightarrow y=\dfrac{255}{51} \\\ & \therefore y=5 \\\ \end{aligned}$
Substituting $y=5$ in equation $(ii)$, we get
$5x-4\left( 5 \right)=-5$
Simplifying this equation and solving it for variable $x$, we get the following,
$\begin{aligned} & 5x-20=-5 \\\ & \Rightarrow 5x=15 \\\ & \therefore x=3 \\\ \end{aligned}$
Therefore, the solution of the given system of linear equations is $\left( 3,5 \right)$.

Note:
There are other methods to solve a system of linear equations. These methods are graphing and Gauss elimination methods. We are asked to write the solution in an ordered pair. An ordered pair is a list of two values in a parentheses in a fixed order. Here, we always write the x-coordinate first and then the y-coordinate.