The substitution and elimination methods involves many arithmetic operations, whereas the cross multiplication method utilize the coefficients effectively, which simplifies the procedure to get the solution.

**Solving by Cross Multiplication Method**

The
substitution and elimination methods involves many arithmetic operations, whereas
the cross multiplication method utilize the coefficients effectively, which simplifies
the procedure to get the solution. This method of cross multiplication is so called
because we draw cross ways between the numbers in the denominators and cross multiply
the coefficients along the arrows ahead. Now let us discuss this method as follows:

Suppose we are given a pair of linear simultaneous equations such as

*a*_{1}* x *+ *b*_{1}* y *+c_{1}
= 0 ...(1)

*a*_{2}* x *+ *b*_{2}* y *+ *c*_{2} = 0 ...(2)

such
that *a*_{1}/*a*_{2} ≠ *b*_{1}/*b*_{2}
. We can solve them as follows :

(1)
× *b*_{2} – (2) × *b*_{1} gives *b*_{2} (*a*_{1}* x *+ *b*_{1}* y *+c_{1} ) − *b*_{1}(*a*_{2}* x *+ *b*_{2}* y *+c_{2} ) = 0

⇒* x *(*a*_{1}*b*_{2} − *a*_{2}*b*_{1}
) = (*b*_{1}c_{2} − b_{2}c_{1})

(1) × *a*_{2}
– (2) × *a*_{1} similarly can be
considered and that will simplify to

* y *=
(c_{1}*a*_{2} −* c*_{2}*a*_{1}) / (a_{1}*b*_{2}* *– a_{2}*b*_{2}* *)

Hence the solution for the system is

Solve
3*x* − 4 *y* = 10 and 4*x* +
3*y* = 5 by the method of cross multiplication.

The
given system of equations are

3*x*
−
4 *y* = 10 ⇒
3*x* − 4 *y* −10 = 0 .....(1)

4*x*
+
3*y* = 5 ⇒
4*x* + 3*y* − 5 = 0 .....(2)

For
the cross multiplication method, we write the co-efficients as

Thus
the solution is *x* = 2, *y* = –1.

**Example 3.53**

Solve
by cross multiplication method : 3*x* + 5*y* =
21; −7
*x* − 6 *y* = −49

*Solution*

The
given system of equations are 3*x* + 5*y* −
21 =
0; −7
*x* − 6 *y* + 49 = 0

Now
using the coefficients for cross multiplication, we get,

Here
*y* /0 = 1/17 is to mean *y* = 0/17 . Thus, *y* /0 is only a notation
and it is not division by zero. It is always true that division by zero is not defined.

Tags : Solving simultaneous linear equations in Two Variables | Example Solved Problems | Algebra | Maths , 9th Maths : UNIT 3 : Algebra

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9th Maths : UNIT 3 : Algebra : Solving by Cross Multiplication Method | Solving simultaneous linear equations in Two Variables | Example Solved Problems | Algebra | Maths

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