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1
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- Systems of Linear Equations
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2
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- A collection of two, three or
more equations
- is defined as a system of equations.
- Example: x + 5 = y or ax + by = e
- 2x – 2y = 3 cx + dx =
f
- Where not all of a, b, c, and d equal 0, the graph of each equation in this
system is a straight line.
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3
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- In a geometric sense, because
the graphs are straight
lines, we are confronted with three possibilities:
- y y y
- x x x
- (a) (b) (c)
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- In case (a) the two lines coincide.
We say that the equations in the system are dependent.
- In case (b) the lines are parallel and do not intersect. We say that the equations in the
system are inconsistent.
- In case (c) the lines intersect
at exactly one and only one point.
We say that the system of equations is consistent.
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5
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- Example: 2x + 3y = 1
- 3x – y = 7
- Solving the second equation
gives y = 3x – 7. Substituting
this expression for y into the first equation yields 2x + 3(3x – 7) = 1
- 2x + 9x –
21 = 1
- 11x
– 21 = 1
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11x = 22
-
x = 2
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6
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- Next, substitute the value x = 2
into either of the original equations to obtain the value of y.
- 2x + 3y = 1 3x - y = 7
- 2(2) + 3y = 1 3(2) - y = 7
- 3y = 1 –
4 6 - y =
7
- y = - - y =
7 - 6
- y = - 1 y = - 1
- The solution of the system of
linear equations is (2, –1).
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7
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- Example: x – y = – 2 (1)
- 2x – 3y = – 7 (2)
- Step 1 Multiply equation (1) by
–3 –3x + 3y = 6 (1´)
- Step 2 Add equation (1´) to
equation (2)
- –3x + 3y
= 6 (1´)
- 2x + (–3y) = –7 (2)
- – x = –1
or equivalently, x = 1
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8
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- Step 3 Substitute the value x = 1
into either of the
- original equations to
obtain the value of y.
- x – y = –2 2x - 3y = -7
- 1 – y = –2 2(1) - 3y = -7
- –y = –3 2 - 3y = -7
- y = 3 -3y = -9
-
y = 3
- The solution of the system of linear equations is (1, 3).
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Notes