Add the following algebraic expression using both horizontal and vertical methods. Did you get the same answer with both methods?
${x^2} - 2xy + 3{y^2}; 5{y^2} + 3xy - 6{x^2}$

Hint – In this question we are given two different equations and we need to add them horizontally and vertically. Horizontally addition means we simply need to add the two equations marked in horizontal fashion, normally as done during equation addition. Vertical method addition means the equation needs to be added in vertical alignment such that the left hand side of one equation falls in the left hand side of the other whereas the right hand side of one equation falls under the right hand side of the other equation.
Complete step by step answer:
Given algebraic expression are
${x^2} - 2xy + 3{y^2}; 5{y^2} + 3xy - 6{x^2}$
Now we have to add these following expressions using horizontal and vertical methods.
$\left( i \right)$ Horizontal method
In this method, all expressions are written in a horizontal line and then the terms are arranged to collect all the groups of like terms and then added.
So first equation is
$ \Rightarrow {x^2} - 2xy + 3{y^2}$………………………….. (1)
Second equation is
$ \Rightarrow 5{y^2} + 3xy - 6{x^2}$
The above equation is also written as
$ \Rightarrow - 6{x^2} + 3xy + 5{y^2}$…………………… (2)
Now add these two equations horizontally we have,
$ \Rightarrow \left( {{x^2} - 2xy + 3{y^2}} \right) + \left( { - 6{x^2} + 3xy + 5{y^2}} \right)$
Now collect like terms we have,
\[ \Rightarrow \left( {1 - 6} \right){x^2} + \left( { - 2 + 3} \right)xy + \left( {3 + 5} \right){y^2}\]
Now simplify we have,
$ \Rightarrow - 5{x^2} + xy + 8{y^2}$………………………. (3)
So, this is the required addition using the horizontal method.
$\left( {ii} \right)$ Vertical method
Vertical addition is a method of adding where you place the numbers vertically, top to bottom, and line up the numbers with the same place values in the same columns. This allows you to add the numbers in each place value separately to come up with the answer.
So first equation is
$ \Rightarrow {x^2} - 2xy + 3{y^2}$………………………….. (1)
Second equation is
$ \Rightarrow 5{y^2} + 3xy - 6{x^2}$
The above equation is also written as
$ \Rightarrow - 6{x^2} + 3xy + 5{y^2}$…………………… (2)
Now add these two equations vertically we have,
\[
  \;{\text{ }}\left( {{x^2} - 2xy + 3{y^2}} \right) \\
  {\text{ + }}\left( { - 6{x^2} + 3xy + 5{y^2}} \right) \\
  {\text{ }}\overline {\left( {1 - 6} \right){x^2} + \left( { - 2 + 3} \right)xy + \left( {3 + 5} \right){y^2}} \\
\]
Now simplify we have,
$ \Rightarrow - 5{x^2} + xy + 8{y^2}$……………………….. (4)
So, this is the required addition using the vertical method.
So as we see that both the equations (3) and (4) are the same so we get the same answer using both horizontal and vertical methods.

Note – Whenever we face such types of problems the key concept is simply to add the given equations using the methods which are being asked in the problem. Remember that during equation addition or even subtraction the coefficients of same variables get added or subtracted. Use this concept to get on the right track to reach the answer.