Some problems are presented below as warm-up exercises:
Question 1:
Solve : and
Question 2:
Solve: and
Question 3:
Solve: and
Question 4:
Solve : and
Some other standard set of systems of equations and techniques for solving them are presented below:
Any pair of equations of the form
….Equation I
…Equation II
where p is any numerical quantity, can be reduced to one of the cases in the warm-up exercises; for, by squaring II and combining with I, an equation to find xy is obtained, the solution can then be completed by the aid of equation II.
Example 1:
Solve …Equation 1
and …Equation 2.
By division, …Equation 3.
From Equation 2, ;
by subtraction, and
…Equation 4.
From Equation (2) and (4), or
Example 2:
Solve: …Equation I
and …Equation 2
Dividing (1) by (2), …Equation 3.
From Equation (2) and (3), by addition, ; by subtraction
, hence, we get the following:
, or
Example 3:
Solve: …Equation (1)
and …Equation (2)
From (1), by squaring, ,
and by subtraction,
adding to (2),
and hence, .
Combining with (1), , or
;
, or
; so, the solutions are:
and
.
The following method of solution may always be used when the equations are of the same degree and homogeneous.
Example:
Solve …Equation 1
and …Equation 2.
Put , and substitute in both equations. Thus,
…Equation 3.
…Equation 4.
By division,
Hence, ; or
; hence,
; so
or
.
Case (i):
Take and substitute in either (3) or (4).
From (3):
, so
; so
.
Case (ii):
Take , then from (3)
, so we get
, so
; and hence,
.
When one of the equations is of first degree and the other of a higher degree, we may from the simple equation, find the value of one of the unknowns in terms of the other, and substitute in the second equation.
Example:
Solve …Equation (1)
and …Equation (2)
From (1), we have
and substituting in (2),
Hence, , and
.
The examples we have presented above will be sufficient as a general explanation of the methods to be employed; but, in some cases, special artifices are necessary.
Example:
Solve: …Equation (1)
and …Equation (2)
From (1) and (2), we have ;
that is, ;
or, , hence,
Case:
Combining with (2), we obtain
, so
; and by substituting in
, we get
.
Case:
Combining with (2), we obtain
; hence,
, and
Example:
Solve: …Equation (1)
and …Equation (2)
From (1), we have ;
From (2), we have ;
By subtraction, ; that is,
; so
.
Case:
Substituting in (2), gives
. From these equations, we obtain
or
;
, or
.
Case:
Substituting in (2), gives
. From these equations, we obtain
, and
.
Hope you enjoyed it,
Nalin Pithwa.