Question

If 2x + y = 23 & 4x - y = 19; find the values of x - 3y & 5y - 2x.
A) The values of \[x - 3y\& 5y - 2x\] are -20 and 31 respectively.
B) The values of \[x - 3y\& 5y - 2x\] are 0 and 3 respectively.
C) The values of \[x - 3y\& 5y - 2x\] are 14 and -9 respectively.
D) The values of \[x - 3y\& 5y - 2x\] are 5 and 23 respectively.

Answer 1

Hint: First find the values of x and y from \[2x + y = 23\& 4x - y = 19;\] and then use those values to find the values of \[x - 3y\& 5y - 2x\] . You can use any technique to solve linear equations, elimination or substitution is preferred.

Complete step by step answer:
Let us try to find the values of x and y by using \[2x + y = 23\& 4x - y = 19\] these equations from the elimination method.
For elimination methods at least one variable must share the same coefficient.
It is clearly seen that coefficient of y is same in both the equation so lets start doing it by adding the equations as y will cancel out
Let us assume that
\[\begin{array}{l}
2x + y = 23......................................(i)\\
4x - y = 19......................................(ii)
\end{array}\]
Adding (i) and (ii) we will get
\[\begin{array}{l}
 \Rightarrow (2x + y) + (4x - y) = 23 + 19\\
 \Rightarrow 2x + y + 4x - y = 23 + 19\\
 \Rightarrow 6x = 42\\
 \Rightarrow x = \dfrac{{42}}{6}\\
 \Rightarrow x = 7
\end{array}\]
Now as we have the value of x let us put it in equation (i) to get the value of y
\[\begin{array}{l}
\therefore 2x + y = 23\\
 \Rightarrow 2 \times 7 + y = 23\\
 \Rightarrow 14 + y = 23\\
 \Rightarrow y = 23 - 14\\
 \Rightarrow y = 9
\end{array}\]
So now we have the value of x and y as 7 and 9 respectively.
As now we have the values of x and y let us put it in \[x - 3y\& 5y - 2x\]
For \[x - 3y\] we will have
\[\begin{array}{l}
 = 7 - 3 \times 9\\
 = 7 - 27\\
 = - 20
\end{array}\]
For \[5y - 2x\] we will have
\[\begin{array}{l}
 = 5 \times 9 - 2 \times 7\\
 = 45 - 14\\
 = 31
\end{array}\]

So, the correct answer is “Option A”.

Note: We can also do this question by using substitution method that is we have to bring one either x or y in terms of another using one equation and them substitute it in the other one as a result the equation thus formed will have only one variable from where we can get either one of the value and then using that value we can get the other one.