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Simultaneous
Linear Equations
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System
of linear equations word problems
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Solving
systems of equations graphically
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Independent
equations, inconsistent equations and dependent equations
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System
of linear equations word problems
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Example: A two-digit number enlarges by nine when its digits reverse. The
same two-digit number divided by sum of its digits gives quotient 5
and reminder 4. Find the two-digit number.
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Solution:
Let
x be ten's digit and
y
units' digit, then 10x
+
y is a two-digit
number.
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Then, written are conditions of the given problem:
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Example: An isosceles triangle with the sides 6 cm longer then base has the
perimeter 48 cm.
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What is the length of its base and the sides?
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Solution:
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Example: The perimeter of a rectangle is 42 cm. The ratio between its width
and the length is 3 : 4.
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Find the length and width of the rectangle.
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Solution:
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Solving
systems of equations graphically
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The system of two equations in two unknowns can be represented
graphically in the Cartesian plane as two lines l1
and l2.
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l1 ::
a1x+ b1y
+ c1 = 0 |
l2 ::
a2x
+ b2y
+ c2 = 0 |
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The coordinates (x,
y)
of the intersection of the two lines are the
values of the variables that make both equations true.
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Independent
equations, inconsistent equations and dependent equations
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There are three possibilities:
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a) Independent equations, the lines intersect
in one point. There is a unique solution if |
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b)
Inconsistent equations, the lines are parallel but distinct (have the same slope), that is |
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and there is
no solution. |
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c)
Dependent equations, equations describe the same line, that is |
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thus, there are an
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infinite number of solutions to the system.
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Example:
Solve graphically given system of
linear equations:
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Solution:
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l1 ::
2x
+ 3y -
4 = 0 |
l2 ::
-x
+ 2y -
5 = 0 |
Coefficients
satisfy the condition: |
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so, the
lines intersect. |
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Solve equation y
= 0 to get the
x-intercept, and
x
= 0 to get the
y-intercept.
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Thus, obtained are the points, (2,
0) and
(0,
4/3) of
the line l1, and
(-5,
0) and
(0,
5/2) of
the line l2.
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The
lines intersect at (-1,
2).
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College
algebra contents C
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