Question

Solve the following equations: $3x-y=7, x+4y=11$

Answer 1

Hint: In the question we are given two linear equations of two variables. First we will learn the names of methods which we can use to solve these equations. Then we will use the substitution method to solve these as it is the easiest method to get our unique solution and then we will also see trivial solutions for these equations.

Complete step-by-step solution:
In the question we are given two equations
$3x-y=7;x+4y=11$
Let us name the equations as $eq(1)$ and $eq(2)$ .
$3x-y=7......eq(1)$
$x+4y=11......eq(2)$
We will use the substitution method to solve these equations.
We will take the value of $y$ from $eq(1)$ and then we will substitute it in $eq(2)$ .
Transposing $y$ to the right hand side and $7$ to the left hand side
$\begin{align}
  & 3x-y=7 \\
 & \Rightarrow 3x-7=y \\
\end{align}$
We can write it as
$\Rightarrow y=3x-7......eq(3)$
Using this in $eq(2)$
$\begin{align}
  & x+4y=11 \\
 & \Rightarrow x+4(3x-7)=11 \\
\end{align}$
Solving the equation we will get
$\begin{align}
  & x+4(3x-7)=11 \\
 & \Rightarrow x+12x-28=11 \\
 & \Rightarrow 13x-28=11 \\
\end{align}$
Transposing $28$ to right hand side
$\begin{align}
  & 13x-28=11 \\
 & \Rightarrow 13x=28+11 \\
 & \Rightarrow 13x=39 \\
\end{align}$
Transposing $13$ to right hand side
$\begin{align}
  & 13x=39 \\
 & \Rightarrow x=\dfrac{39}{13} \\
\end{align}$
Solving this we will get
$x=3$
Now substituting this in $eq(3)$
$\begin{align}
  & y=3x-7 \\
 & \Rightarrow y=3\times 3-7 \\
\end{align}$
Simplifying right hand side
$\begin{align}
  & y=9-7 \\
 & \Rightarrow y=2 \\
\end{align}$
$\therefore x=3$ And $y=2$ is our required answer.

Additional information:
Linear equation in two variables is a mathematical statement which has two variables of degree one.
In general a linear equation of two variables is represented as
$ax+by+c=0$
In which $x$ and $y$ are variables.
$a$ Is the coefficient of $x$.
$b$ Is the coefficient of $y$ .
$c$ Is constant.
To solve linear equations in two variables we need at least two equations of the same variables.
Now we will learn methods to solve linear equations of two variables.
To solve linear equations in two variables we have four methods
a) Graphical Method
b) Substitution Method
c) Elimination Method
d) Cross Multiplication Method.
We can use any method to solve equations.
In substitution we take out the value of one variable in terms of another variable from one equation then we will substitute this value in another equation, then it will become a linear equation in one variable and we can solve it using the transposition method. From this we will get the value of one variable and then we will use this value to find out the value of another variable.

Note: To solve a linear equation in two variables we will need at least two equations. If only one equation of two variables is given then there will be infinite solutions and there will be no unique answer. In general we can say that to solve linear equations of $n$ variables we will at least need $n$ number of equations.