1. Systems of Linear Equations
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Systems of Linear Equations

2. Systems of Linear Equations
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 = 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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Systems of Linear Equations

3. Graphs of Linear Systems
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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Systems of Linear Equations

4. Systems of Linear Equations
Graphs of Linear Systems
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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Systems of Linear Equations

5. Solving Linear Equations by the Substitution Method (part 1)
Example: x + 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
11x = 22
x = 2
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Systems of Linear Equations

6. Solving Linear Equations by the Substitution Method (part 2)
Next, substitute the value x = 2 into either of the original equations to obtain the value of y.
2x + 3y = x - y = 7
2(2) + 3y = (2) - y = 7
3y = 1 – y = 7
y = y = 7 - 6
y = y = - 1
The solution of the system of linear equations is (2, –1).
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Systems of Linear Equations

7. Solving Linear Equations by the Elimination Method (part 1)
Example: x – y = – (1)
2x – 3y = – 7 (2)
Step 1 Multiply equation (1) by – –3x + 3y = 6 (1´)
Step 2 Add equation (1´) to equation (2)
–3x y = (1´)
2x + (–3y) = – (2)
– x = –1 or equivalently, x = 1
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Systems of Linear Equations

8. Solving Linear Equations by the Elimination Method (part 2)
Step 3 Substitute the value x = 1 into either of the
original equations to obtain the value of y.
x – y = – x - 3y = -7
1 – y = – (1) - 3y = -7
–y = – y = -7
y = y = -9
y = 3
The solution of the system of linear equations is (1, 3).
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Systems of Linear Equations