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Algebra
Clement
Radcliffe, Contributor
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| Students
of Pentab High School, downtown
Kingston, get attention from a
teacher. - Ian Allen Photo |
WE
WILL continue algebra by reviewing aspects
of inequations.
POINTS
TO NOTE
*
Equations identify the relationship
between variables (for example, 2x
+ y = 3) or indentify the value of
variables (for example, x = 7).
*
Inequations are similar except that
instead of equality, the following
relationships are
considered.
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Less than < * Less than or equal
to ¾
*
Greater than > * Greater than or
equal to >=
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The solution of an Inequation is the
domain of x, that is the set of values
of x which satisfies the inequation.
LINEAR
INEQUATIONS
Examples:
*x
> 2 *x + y < 6
The
above Linear Inequations may be represented
graphically as follows:
This
range includes all values of x greater
than 2. Example, x = 4.
This range includes all values of
x and y which, when added are less
than or equal to 6. Example, x = 1,
y =3.
SIMPLIFICATION
OF LINEAR INEQUATIONS
POINTS
TO NOTE
*
The procedure is similar to the solution
of a linear equation.
Given
the Inequation 2x - 1 >3
ie.
2x > 3 + 1
ie.
x > 2
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Where both sides of the Inequation
are divided by a negative number,
then the inequality sign is reversed.
Example:
Solve 2 - 3x < - 1
ie.
-3x < - 1 -2 ... -3x < - 3,
Dividing by -3.
ie.
x >1
Please
attempt the following practice examples
Find
the domain of x for which:
1.
6x -3 < 7
2.
1 - 2x > 4
SIMULTANEOUS
LINEAR EQUATIONS
*
The solution of the simultaneous equations
is the pair of x and y values which
satisfy both equations.
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If both equation are plotted on a
graph, it is the point of intersection
of both lines.
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You may use the elimination or substitution
method. However, the former is recommended.
EXAMPLE
1.
Solve
the simultaneous equations:
2x
- y = -1 ....... (1)
3x
-y = 2 ........ (2)
Subtracting
equation (2) from (1)
-
x = -3 ... x = 3
Substituting
x = 3 into (1)
ie.
6 - y = -1
ie.
y = 7
Answer
is x = 3, y = 7.
You
may substitute the answer x = 3 and
y =7 into both equations in order
to check your answer.
Do
you realise that since the coefficient
of y is -1 in both equations, you
eliminate y by subtracting. If the
coefficients differ in sign ONLY,
that is, if the coefficients of y
are -1 and +1, then you eliminate
by adding.
EXAMPLE
2
Solve
the simultaneous equation:
5x
+ 3y = 31 ........ (1)
2x
+ y =12 ......... (2)
Multiply
equation (2) by 3 and then subtract
equation (1) from equation (3).
6x
+ 3y =36 ......... (3)
5x
+ 3y =31 ......... (1)
ie.
x =5
Substituting
x = 5 in (2)
10
+ y = 12 ........ (2)
The
following is an example of the substitution
method:
EXAMPLE
3
Solve
the simultaneous equations:
5x
+ 3y =31
2x
+y =12
5x
+ 3y =31 ........... (1)
2x
+y =12............ (2)
From
equation (2), y = 12 - 2x
Substituting
into (1)
ie.
5x + 3(12- 2x) =31
5x
+ 36 - 6x =31
ie.
- x =31 - 36
ie.
-x = -5 or x = 5
Substituting
into equation (2)
ie.
10 + y =12
ie.
y = 2
ie .
Answer is: x = 5 and y + 2
Please
attempt to solve the following simultaneous
equation:
a)
2x - 2y = 1 b) x + y = 7
7x
= 2y = 17 2x + y = 10
c)
x - y = -5 d) 3x -2y =7
3x
+ 2y = -5 -x + 3y = -7
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Clement Radcliffe is principal
of Glenmuir High School in Clarendon.
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