# LINEAR EQUATIONS

# LINEAR EQUATION

**HOMEWORK **1.1: 8,14,20,24,46,26*,36* (Problems with * are
recommended only and are not turned in).

** SYSTEM OF LINEAR EQUATIONS . **A collection of linear
equations is called a system of linear equations.

An example is

This system consists of three equations for three unknowns
x, y, z. Linear means that no nonlinear terms like

x^{2}, x^{3}, xy, yz^{3}, sin(x) etc. appear. A formal definition of linearity will be
given later.

**LINEAR EQUATION. **The equation ax+by = c is the general
linear equation in two variables and ax+by+cz =

d is the general linear equation in three variables. The general linear equation
in n variables has the form

a_{1}x_{1} + a_{2}x_{2} + ... + a_{n}x_{n} = a_{0} . Finitely many of such equations form a system of
linear equations.

** SOLVING BY ELIMINATION .**

Eliminate variables . In the first example, the first equation gives z = 3x − y.
Substituting this into the

second and third equation gives

or

The first equation leads to y = 4/3x and plugging this
into the other equation gives 8x− 16/3x = 9 or 8x = 27

which means x = 27/8. The other values y = 9/2, z = 45/8 can now be obtained.

**SOLVE BY SUITABLE SUBTRACTION .**

Addition of equations . If we subtract the third equation from the second, we get
3y − 4z = −9 and add

three times the second equation to the first, we get 5y − 4z = 0. Subtracting
this equation to the previous one

gives −2y = −9 or y = 2/9.

**SOLVE BY COMPUTER.**

Use the computer. In Mathematica:

Solve[{3x − y − z == 0,−x + 2y − z == 0,−x − y + 3z == 9}, {x, y, z}] .

But what did Mathematica do to solve this equation? We will look in this course at some efficient algorithms.

**GEOMETRIC SOLUTION.**

Each of the three equations represents a plane in

three-dimensional space. Points on the first plane

satisfy the first equation. The second plane is the

solution set to the second equation. To satisfy the

first two equations means to be on the intersection

of these two planes which is here a line. To satisfy

all three equations, we have to intersect the line with

the plane representing the third equation which is a

point.

**LINES, PLANES, HYPERPLANES.**

The set of points in the plane satisfying ax + by = c form a** line.**

The set of points in space satisfying ax + by + cd = d form a **plane.**

The set of points in n-dimensional space satisfying a_{1}x_{1} + ... + a_{n}x_{n} = a_{0}
define a set called a hyperplane.

**RIDDLES:**

”25 kids have bicycles or tricycles. Together they

count 57 wheels. How many have bicycles?”

**Solution.** With x bicycles and y tricycles, then x+

y = 25, 2x + 3y = 57. The solution is x = 18, y = 7.

”Tom, the brother of Carry has twice as many sisters

as brothers while Carry has equal number of sisters

and brothers. How many kids is there in total in this

family?”

**Solution** If there are x brothers and y sisters, then

Tom has y sisters and x−1 brothers while Carry has

x brothers and y − 1 sisters. We know y = 2(x −

1), x = y − 1 so that x + 1 = 2(x − 1) and so x =

3, y = 4.

**INTERPOLATION.**

Find the equation of the

parabola which passes through

the points P = (0,−1),

Q = (1, 4) and R = (2, 13).

**Solution.** Assume the parabola con-

sists of the set of points (x, y) which

satisfy the equation ax^2 + bx + c = y.

So, c = −1, a+ b + c = 4, 4a+ 2b + c =

13. Elimination of c gives a + b =

5, 4a + 2b = 14 so that 2b = 6 and

b = 3, a = 2. The parabola has the

equation 2x^2 + 3x − 1 = 0

**TOMOGRAPHY**

Here is a toy example of a problem one has to solve for magnetic

resonance imaging (MRI). This technique makes use of the ab-

sorbtion and emission of energy in the radio frequency range of

the electromagnetic spectrum.

Assume we have 4 hydrogen atoms, whose nuclei are excited
with

energy intensity a, b, c, d. We measure the spin echo in 4 different

directions. 3 = a + b,7 = c + d,5 = a + c and 5 = b + d. What

is a, b, c, d? Solution: a = 2, b = 1, c = 3, d = 4. However,

also a = 0, b = 3, c = 5, d = 2 solves the problem. This system

has not a unique solution even so there are 4 equations and 4

unknowns .

**INCONSISTENT.** x − y = 4, y + z = 5, x + z = 6 is a system
with no solutions. It is called inconsistent.

**EQUILIBRIUM.** As an example of a system with many

variables, consider a drum modeled by a fine net. The

heights at each interior node needs the average the

heights of the 4 neighboring nodes. The height at the

boundary is fixed. With n^2 nodes in the interior, we

have to solve a system of n^2 equations. For exam-

ple, for n = 2 (see left), the n^2 = 4 equations are

To the right, we see the solution to a problem with

n = 300, where the computer had to solve a system

with 90′000 variables.

**LINEAR OR NONLINEAR?**

a) The ideal gas law PV = nKT for the P, V, T , the pressure p, volume V and
temperature T of a gas.

b) The Hook law F = k(x − a) relates the force F pulling a string extended to
length x.

c) Einsteins mass-energy equation E = mc^2 relates restmass m with the energy E
of a body.

**ON THE HISTORY.** In 2000 BC the Babylonians already studied problems which led to
linear equations.

In 200 BC, the Chinese used a method similar to Gaussian elimination to solve
systems of linear equations.

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