Question

Solve the following pair of equations by cross-multiplication method,
\[(2x-5y=4)\] , \[(3x-8y=5)\]

Answer 1

Hint: Shift the constant number to the LHS of these two equations. Assume two equations \[ax+by+c=0\] and \[dx+ey+f=0\] . Now, compare these two equations with \[ax+by+c=0\] and \[dx+ey+f=0\] . After comparing get the values of a, b, c, d, e, and f in the formula, \[\dfrac{x}{bf-ec}=\dfrac{y}{cd-af}=\dfrac{1}{ae-bd}\] . Now, solve it further and get the values of x and y.

Complete step-by-step answer:

According to the question, we have two equations.
\[2x-5y=4\] ………………..(1)
\[3x-8y=5\] …………….(2)
First of all, make the given equation of the form \[ax+by+c=0\] .
Shifting the constant number 4 to the LHS of equation (1), we get
\[2x-5y-4=0\] …………….(3)
\[3x-8y-5=0\] ……………….(4)
We know the procedure to find the value of x and y using the cross multiplication method.
Let us assume two standard equation which are,
\[ax+by+c=0\] …………….(5)
\[dx+ey+f=0\] ……………(6)
Using the cross multiplication method, we can write
\[\dfrac{x}{bf-ec}=\dfrac{y}{cd-af}=\dfrac{1}{ae-bd}\] …………….(7)
Now, comparing equation (3) with equation (5), we get
\[2x-5y-4=0\]
\[ax+by+c=0\]
Here, we have a=2, b=-5, and c=-4.
Comparing equation (4) with equation (6), we get
\[3x-8y-5=0\]
\[dx+ey+f=0\]
Here, we have d=3, e=-8, and f=-5.
Now, we have to put the values of a, b, c, d, e, and f in equation (7).
Putting the values of a, b, c, d, e, and f in equation (7), we get
\[\begin{align}
  & \dfrac{x}{bf-ec}=\dfrac{y}{cd-af}=\dfrac{1}{ae-bd} \\
 & \dfrac{x}{(-5)(-5)-(-8)(-4)}=\dfrac{y}{(-4)(3)-(-5)(2)}=\dfrac{1}{(2)(-8)-(3)(-5)} \\
\end{align}\]
On solving we get,
\[\dfrac{x}{(-5)(-5)-(-8)(-4)}=\dfrac{y}{(-4)(3)-(-5)(2)}=\dfrac{1}{(2)(-8)-(3)(-5)}\]
\[\dfrac{x}{25-32}=\dfrac{y}{-12-(-10)}=\dfrac{1}{(-16)-(-15)}\]
\[\dfrac{x}{-7}=\dfrac{y}{-12+10}=\dfrac{1}{(-16)+15}\]
\[\dfrac{x}{-7}=\dfrac{y}{-2}=\dfrac{1}{-1}\]
We now have two equations to solve. One is \[\dfrac{x}{-7}=-1\] and the other is \[\dfrac{y}{-2}=-1\] .
\[\dfrac{x}{-7}=-1\] ……………….(8)
\[\dfrac{y}{-2}=-1\] ……………….(9)
Now, solving equation (8), we get
\[\dfrac{x}{-7}=-1\]
\[\begin{align}
  & \Rightarrow x=-1(-7) \\
 & \Rightarrow x=7 \\
\end{align}\]
Now, solving equation (9), we get
\[\dfrac{y}{-2}=-1\]
\[\begin{align}
  & \Rightarrow y=(-2)(-1) \\
 & \Rightarrow y=2 \\
\end{align}\]
Hence, the value of x and y is 7 and 2 respectively.

Note: In this question, one may do a silly mistake in the formula and put the values of a, b, c, d, e, and f in \[\dfrac{x}{bf-ec}=\dfrac{y}{af-cd}=\dfrac{1}{ae-bd}\] . This is wrong. This is not the correct formula where the values of a, b, c, d, e, and f should be placed. So, we have to keep the formula in mind.