60 MATHEMATICS
3 – 46^23
(^5) – 74 35
x y 1
Note that you can write the solution given by Equations (5) and (6) in the
following form :
12 21 12 21 12 21
xy 1
bc b c ca c a ab a b✁ ✁ ✁ (8)
In remembering the above result, the following diagram may be helpful to
you :
The arrows between the two numbers indicate that they are to be multiplied and
the second product is to be subtracted from the first.
For solving a pair of linear equations by this method, we will follow the following
steps :
Step 1 : Write the given equations in the form (1) and (2).
Step 2 : Taking the help of the diagram above, write Equations as given in (8).
Step 3 : Find x and y, provided a 1 b 2 – a 2 b 1 ✂ 0
Step 2 above gives you an indication of why this method is called the
cross-multiplication method.
Example 14 : From a bus stand in Bangalore , if we buy 2 tickets to Malleswaram and
3 tickets to Yeshwanthpur, the total cost is Rs 46; but if we buy 3 tickets to Malleswaram
and 5 tickets to Yeshwanthpur the total cost is Rs 74. Find the fares from the bus stand
to Malleswaram, and to Yeshwanthpur.
Solution : Let Rs x be the fare from the bus stand in Bangalore to Malleswaram, and
Rs y to Yeshwanthpur. From the given information, we have
2 x + 3y = 46, i.e., 2x + 3y – 46 = 0 (1)
3 x + 5y = 74, i.e., 3x + 5y – 74 = 0 (2)To solve the equations by the cross-multiplication method, we draw the diagram as
given below.
b 1 c 1 a 1 b 1b 2 c 2 a 2 b 2x y 1