Question

How do you solve the system $x-y=11$ and $2x+y=19$ ?

Answer 1

Hint: The two equations are solved by linearly combining them into a single equation in such a way as to eliminate one variable from the two equations. The value of the other variable having been known; we can determine the value of the other variable.

Complete step by step solution:
The given equations that we have at our disposal are,
$x-y=11$
$2x+y=19$
We can solve the problem by taking a linear combination of the two equations. By linear combination, we mean that if two expressions or equations are to be linearly combined, then we write it in the form $aX+bY$ where, X and Y are the expressions or equations and, a and b are two multiplying factors. During this linear combination, we can eliminate one of the two variables. The resulting combined equation has thus become an equation of a single variable. From this equation, we can find out the value of this variable. By substituting this value of the found-out variable into one of the given equations, we can find out the value of the other variable. In this way, we can solve the two given equations.
In the given problem, if we closely observe the two equations, we can see that if we simply add the two equations or in other words, take a linear combination of them with both the factors as $1$ , we can eliminate y.
$\begin{align}
  & \left( x-y \right)+\left( 2x+y \right)=11+19 \\
 & \Rightarrow 3x=30 \\
 & \Rightarrow x=10 \\
\end{align}$
Putting this value of x in the first equation, we get,
$\begin{align}
  & \Rightarrow 10-y=11 \\
 & \Rightarrow y=-1 \\
\end{align}$
Therefore, we can conclude that the solution of the given equations is $x=10,y=-1$ .

Note: We should be careful while linearly combining the two equations especially when including factors, as students often forget to multiply the factor to the right-hand side of the equation. This problem can also be solved using graphs by locating the point of intersection of the two lines.