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Systems of Linear Equations

Introduction: One equation in one variable

To solve the equation

3x + 2 = 14

3x + 2 − 2 = 14 − 2

3x = 12

x = 4

we:
•  Subtract two from each side to have the variable on one side
and the constant on the other
Divide both sides by 3 to scale the coefficient of x to be one .
• See the solution x = 4

Alegebra (al-jabr) = “Restoration”
= “Do the same thing to both sides of the equation”

Systems of two equations in two variables

To solve the system

x + 2y = 3
x + 5y = −3

We might
Subtract both equations to get −3y = 6
Divide by −3 to get y = −2
Substitute back into the first equation to get

x + 2(−2) = 3 -> x = 7

So the solution is the pair (7,−2).

Geometric interpretation

Each equation represents a line in the plane

Anamolies

Two lines need not intersect in a point!
• They may be parallel
• They may be coincidental

Systems of three equations in three variables

Solve
x − 2y + z = 0
2y − 8z = 8
−4x + 5y + 9z = −9

Solution
x = 29, y = 16, z = 3

Geometric Interpretations

• In three variables, the solution set to one equation is a plane
Multiple equations give intersections of these planes.
• How can planes intersect?

• not at all
lines
• a point

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