# Systems of Linear Equations

## Introduction: One equation in one variable

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

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

Prev | Next |