Question

How do you solve this system of equations: $x+4y=5$ ; $4x-2y=11$ ?

Answer 1

Hint: There are 2 unknown variables, x and y and 2 equations in the given question. We can solve the equation by eliminating any one of the 2 variables. So, we will multiply the second equation by 2 and then add it with the first equation to eliminate the variable x.

Complete step by step answer:
The given 2 equations in the question
$x+4y=5$ …..eq1
$4x-2y=11$ ……eq2
Let’s solve the question by eliminating y.
In eq2 we can multiply 2 in both LHS and RHS
The equation will be $8x-4y=22$ ….eq3
Now adding eq3 and eq4
$9x=27$
So $x=\dfrac{27}{9}$
$\Rightarrow x=3$
Now we can get the value of y by substituting the value of x in any one of the equations.
Let’s put the value of x in eq1
$3+4y=5$
$\Rightarrow 4y=2$
$\Rightarrow y=\dfrac{1}{2}$
Now we can verify the answers by putting the value of x and y in both the equations.

Note: Another method is to solve by determinant method.
Suppose there are 2 linear equation ${{a}_{1}}x+{{b}_{1}}y={{c}_{1}}$ and ${{a}_{2}}x+{{b}_{2}}y={{c}_{2}}$ the solution to this problem let’s take
D= $\left| \begin{matrix}
   {{a}_{1}} & {{b}_{1}} \\
   {{a}_{2}} & {{b}_{2}} \\
\end{matrix} \right|$
A= $\left| \begin{matrix}
   {{b}_{1}} & {{c}_{1}} \\
   {{b}_{2}} & {{c}_{2}} \\
\end{matrix} \right|$
B= $\left| \begin{matrix}
   {{a}_{1}} & {{c}_{1}} \\
   {{a}_{2}} & {{c}_{2}} \\
\end{matrix} \right|$
$x=-\dfrac{A}{D}$ and $y=\dfrac{B}{D}$ Where $D\ne 0$
If $D=0$ and any one of A and B is not equal to 0 then there will be no solution for the system of equation
If $D=A=B=0$ there will be infinitely many solutions to the system of equations.
If $D\ne 0$ there will be one solution for the system of equations.
In this case ${{a}_{1}}=1,{{b}_{1}}=4,{{c}_{1}}=5$ and ${{a}_{2}}=4,{{b}_{2}}=-2,{{c}_{2}}=11$
Solving the determinant $D=-18,A=54$ and $B=-9$
So $x=-\dfrac{A}{D}$
$x=-\dfrac{54}{-18}$
$\Rightarrow x=3$
And $b=\dfrac{B}{D}=\dfrac{1}{2}$
While solving the system of linear equations we can imagine the equation having 2 unknown as equations of straight in 2-D Cartesian plane so 2 equations means 2 lines will intersect at 1 point unless both lines are parallel or both lines are overlapping. Same goes with 3 unknowns we can imagine the equations as equations of plane in 3D geometry and then solve for the unknown variables.