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.