

Simultaneous
Linear Equations

System
of linear equations word problems

Solving
systems of equations graphically

Independent
equations, inconsistent equations and dependent equations






System
of linear equations word problems

Example: 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:



Example: 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:




Example: 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

The system of two equations in two unknowns can be represented
graphically in the Cartesian plane as two

lines l_{1}
and l_{2}.




l_{1
}::
a_{1}x+ b_{1}y
+ c_{1} = 0 
l_{2
}::
a_{2}x
+ b_{2}y
+ c_{2} = 0 








The coordinates (x,
y)
of the intersection of the two lines are the
values of the variables that make both equations true.


Independent
equations, inconsistent equations and dependent equations

There are three possibilities:

a) Independent equations, the lines intersect
in one point. There is a unique solution if 



b)
Inconsistent equations, the lines are parallel but distinct (have the same slope), that is 

and there is
no solution. 


c)
Dependent equations, equations describe the same line, that is 

thus, there are an

infinite number of solutions to the system.





Example:
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. 






Solve equation y
= 0 to get the
xintercept, and
x
= 0 to get the
yintercept.

Thus, obtained are the points, (2,
0) and
(0,
4/3) of
the line l_{1}, and
(5,
0) and
(0,
5/2) of
the line l_{2}.

The
lines intersect at (1,
2).









Intermediate
algebra contents 



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