
ALGEBRA
 solved problems 






System
of linear equations word problems


107. 
A twodigit number enlarges by nine when its digits reverse. The
same twodigit number divided by


sum of its digits gives quotient 5
and reminder 4. Find the twodigit number.

Solution:
Let
x be ten's digit and
y
units' digit, then 10x+
y is a twodigit
number.

Then, written are conditions of the given problem:



108. 
An isosceles triangle with the sides 6 cm longer then base has the
perimeter 48 cm.


What is the length of its base and the sides?

Solution:




109. 
The perimeter of a rectangle is 42 cm. The ratio between its width
and the length is 3 : 4.


Find the length and width of the rectangle.

Solution:




Solving
systems of equations graphically


110. 
Solve graphically given system of
linear equations.


Solution:

l_{1}
::
2x
+ 3y 
4 = 0 
l_{2 }::
x
+ 2y 
5 = 0 
Coefficients
satisfy the condition: 

so, the
lines intersect. 





Cramer’s
rule (using the determinant) to solve systems of linear equations

Solving system
of two equations in two unknowns
using
Cramer's rule


A system of two equations in two unknowns, the solution to a system
by Cramer’s rule (use of determinants).


the solution to the system 




111. 
Solve given system of
linear equations using Cramer’s rule.


Solution: 




Solving system of three equations in three unknowns
using
Cramer's rule



A determinant of rank
n
can be evaluated by expanding to its cofactors of rank
n 
1, along any row or column taking into account the scheme of
the signs.




For example,
the determinant of rank
n = 3,





112. 
Solve given system of
three equations in three unknowns using method of expanding to
cofactors.


Solution: 



Method of expanding
a determinant
of a rank n
to cofactors

The value of a determinant will not change by adding multiples of
any column or row to any other column or row. This way created are zero entries that simplify subsequent calculations.


113. 
Solve given determinant of the rank four using the method of expanding a determinant to
cofactors.


Solution:



Added is third to the second colon. Then, the second row multiplied
by 3 is added to the first row. The obtained determinant is then
expanded to its cofactors along the second colon:


The first colon multiplied by
1 is added to the third colon. The
obtained determinant is then expanded along the third colon.











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