Section 4-1: System of linear equations in

two variables

Solving a system of linear equations

graphically

In chapter 1 we learned how to solve a linear

equation in one variable, now we have two

equations in two variables, such as:

72

12

yx

yx

To solve this is to find values for x, y such

that they satisfy both equations. We know

that each equation represents a line, so the

solution is exactly the intersection, which is

(3, 5) from the graph below, i.e., x = 3, y =

5.

We see three cases when solving a linear

system graphically:

Solving a system of linear equations by

substitution

Substitution means that you solve for one

variable from one equation and plug into the

other equation. For example: Given

72

12

yx

yx

Solving a system of linear equations using

elimination by addition

Addition means that you multiply each

equation (or only one equation) with a

number then add both equations to get rid of

a variable.

Example 1. Solve by addition: 152

823

yx

yx

Solution:

Example 2. Solve 442

82

yx

yx

Solution:

Note: if you graph these two equations, you

will see two __________lines!

Can you tell directly from the system?

Example 3. Solve 693

462

yx

yx

Solution:

Note: If you graph you see two lines

coincide. In next section we’ll write infinite

number of solutions in the form of

parameters.

How do you tell immediately?

Application Problems:

Example 4 (supply and demand)

Suppose that the supply and demand for

printed baseball caps for a particular

week are 179.1

2.34.0

qp

qp, where p is the

price in dollars and q is the quantity in

hundreds.

a. Find the supply and demand (to the

nearest unit) if baseball caps are $4

each. Discuss the stability of the

baseball cap market at this price level.

b. Find the supply and demand (to the

nearest unit) if baseball caps are $9

each. Discuss the stability of the

baseball cap market at this price level.

c. Find the equilibrium price and

quantity.

d. Graph the two equations in the same

coordinate system and identify the

equilibrium point, supply curve, and

demand curve.

Solution: a. Plug 4 into p in both

equations: 179.14

2.34.04

q

q we get supply q is 2

and demand q is 6.84, since they are in

hundreds, so we have 200 and 684

correspondingly. Since supply quantity is

much less than demand quantity, the

price is going up.

b. Similar to part a we get 1450 for

supply and 421 for demand. Since supply

is much more than demand, the price is

going down.

c. Solve the linear system to get

equilibrium: q = 6 (i.e. 600) and p =

$6.50

d.

Section 4-2: Using augmented matrices to

solve a linear system

A matrix is the form

5

0

1

4

6

2A ,

which is called matrix of size 32 (2 rows

and 3 columns), where the entries

,621 a 013 a .

A square matrix is a matrix with same

number of rows and columns, such as

06

31;

a column matrix is a matrix with only one

column like

2

3

0

;

a row matrix is a matrix with only one row,

as 349 .

In section 4-1 we used addition to solve a

linear system. It works well when we have

two or three variables, but when we have

more than 3 variables, it’s not a very

efficient way. In this section we will use

matrices to do it, and this method works

well for any size of linear system.

Consider the linear system 132

32

yx

yx.

We’ll explain how each step in solving by

addition corresponds to each step in solving

by augmented matrix method.

3 row operations: (‘’ means ‘replace’)

1. Swap two rows: ji RR

2. Multiply a row by a nonzero constant:

ii RkR ( 0k )

3. Multiply a row with a constant and add

it to another row: jji RRcR

Ex1. (a linear system with exactly one

solution)

Solve

2

143

632

21

21

xx

xx

by augmented matrix.