Question 1: Solve the pair of simultaneous equations using the substitution method.
(i)
(ii)
(iii)
(iv)
(v)
Answer:
(i)
The given system of equation is:
… … … … … (i)
… … … … … (ii)
From equation (i), we get
in equation (ii), we get
(ii)
The given system of equation is:
… … … … … (i)
… … … … … (ii)
From equation (ii), we get
in equation (i), we get
(iii)
The given system of equation is:
… … … … … (i)
… … … … … (ii)
From equation (i), we get
in equation (ii), we get
(iv)
The given system of equation is:
… … … … … (i)
… … … … … (ii)
From equation (ii), we get
in equation (i), we get
(v)
The given system of equation is:
… … … … … (i)
… … … … … (ii)
From equation (i), we get
in equation (ii), we get
,
Question 2: Solve the system of equations using the method of elimination.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
(xiii)
(xiv) ,
(xv) ,
(xvi)
(xvii)
Answer:
(i)
… … … … … (i)
… … … … … (ii)
Now multiplying equation (ii) by and subtracting it from equation (i) we get
in (ii) we get
(ii)
If we put in either of the equations, we get
. Similarly if we put
in either of the equations, we get
.
is one of the solutions that satisfies the two equation. However, for
, we follow the following approach.
Dividing each of the given equations by uv we get:
… … … … … (i)
… … … … … (ii)
… … … … … (iii)
… … … … … (iv)
Now multiplying equation( (i) by and equation (ii) by
we get
… … … … … (v)
… … … … … (vi)
Subtracting (vi) from (v) we get
in (iii) we get
(iii)
… … … … … (i)
… … … … … (ii)
Subtracting (ii) from (1) we get
in (i) we get
Therefore
… … … … … (iii)
… … … … … (iv)
Now multiplying equation( (iv) by and equation (iii) by
we get
… … … … … (v)
… … … … … (vi)
Subtracting (vi) from (v) we get
in (v) we get
is the solution for the given system of equations.
(iv)
The given system of equation is:
… … … … … (i)
… … … … … (ii)
Now multiplying equation( (ii) by and equation (i) by
we get
… … … … … (iii)
… … … … … (iv)
Adding (iv) and (iii) we get
in (iii) we get
is the solution for the given system of equations.
(v)
The given system of equation is:
… … … … … (i)
… … … … … (ii)
Now multiplying equation( (i) by and equation (ii) by
we get
… … … … … (iii)
… … … … … (iv)
Subtracting (iv) and (iii) we get
in (i) we get
is the solution for the given system of equations.
(vi)
The given system of equation is:
… … … … … (i)
… … … … … (ii)
Now multiplying equation( (i) by and equation (ii) by
we get
… … … … … (iii)
… … … … … (iv)
Adding (iv) and (iii) we get
in (i) we get
is the solution for the given system of equations.
(vii)
The given system of equation is:
… … … … … (i)
… … … … … (ii)
Now multiplying equation( (i) by and equation (ii) by
we get
… … … … … (iii)
… … … … … (iv)
Adding (iv) and (iii) we get
in (i) we get
is the solution for the given system of equations.
(viii)
… … … … … (i)
… … … … … (ii)
Now multiplying equation (i) by and subtracting (ii) from equation (i) we get
in (ii) we get
(ix)
… … … … … (i)
… … … … … (ii)
Now multiplying equation (ii) by and subtracting it from equation (i) we get
in (ii) we get
… … … … … (iii)
… … … … … (vi)
Adding (iii) and (iv) we get
from (iv)
(x)
… … … … … (i)
… … … … … (ii)
Now multiplying equation (i) by and subtracting (ii) from equation (i) we get
in (ii) we get
… … … … … (iii)
… … … … … (vi)
Adding (iii) and (iv) we get
from (iv)
(xi)
… … … … … (i)
… … … … … (ii)
Now multiplying equation (i) by and subtracting (ii) from equation (i) we get
in (ii) we get
… … … … … (iii)
… … … … … (vi)
Adding (iii) and (iv) we get
from (iv)
(xii)
… … … … … (i)
… … … … … (ii)
Now multiplying equation (ii) by and subtracting (ii) from equation (i) we get
in (ii) we get
… … … … … (iii)
… … … … … (vi)
Adding (iii) and (iv) we get
from (iii)
(xiii)
… … … … … (i)
… … … … … (ii)
Adding (i) and (ii) we get
in (i) we get
(xiv) ,
Given system of equations is
… … … … … (i)
… … … … … (ii)
… … … … … (iii)
From (i)
Substituting this in (ii) and (iii) we get
… … … … … (iv)
… … … … … (v)
Subtracting (v) from (iv) we get
From (v)
(xv) ,
Given system of equations is
… … … … … (i)
… … … … … (ii)
… … … … … (iii)
From (i)
Substituting this in (ii) and (iii) we get
… … … … … (iv)
… … … … … (v)
Adding (v) from (iv) we get
From (v)
(xvi)
… … … … … (i)
… … … … … (ii)
Multiplying (i) by and (ii) by
we get
… … … … … (iii)
… … … … … (iv)
subtracting (iv) from (iii)
in (ii) we get
… … … … … (iii)
… … … … … (vi)
Adding (iii) and (iv) we get
from (iv)
(xvii)
… … … … … (i)
… … … … … (ii)
Adding (i) and (ii)
in (ii) we get
… … … … … (iii)
… … … … … (vi)
Adding (iii) and (iv) we get
from (iv)
Question 3: Solve using the cross multiplication method.
(i)
(ii)
(iii)
(iv) where
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi) where
Answers:
(i)
The given system of equations may be written as
By Cross multiplication, we get
(ii)
The given system of equations may be written as
By Cross multiplication, we get
(iii)
The given system of equations may be written as
By Cross multiplication, we get
(iv) where
Let
The given system of equations may be written as
By Cross multiplication, we get
(v)
The given system of equations is
By Cross multiplication, we get
(vi)
The given system of equations may be written as
By Cross multiplication, we get
(vii)
The given system of equations may be written as
By Cross multiplication, we get
(viii)
We can simplify the equation first:
The given system of equations may be written as
By Cross multiplication, we get
(ix)
First simplify the equation:
The given system of equations may be written as
By Cross multiplication, we get
(x)
First simplify the equations:
Also
By Cross multiplication, we get
(xi) where
Let
Therefore the given set of equations can be written as:
By Cross multiplication, we get