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Chapter 5.1
Systems of Linear Equations
Linear Systems
The definition of a linear equation given in
Chapter 1 can be extended to more
variables; any equation of the form
a1 x1  a2 x2    an xn  b
for real numbers a1 , a2 ,, an
(not all of which are 0) and b, is a linear
equation or a first-degree equation in n
unknowns.
A set of equations is called a system of
equations.
The solutions of a system of equations must
satisfy every equation in the system. If all the
equations in a system are linear, the system is a
system of liner equations, or a linear system.
The solution set of a linear equation in two
unknown (or variables) is an infinite set of
ordered pairs.
Since the graph of such an equation is a
straight line, there are three possibilities for
the solution of a system of two linear
equations in two unknowns, as shown in the
figure.
Consistent
Inconsistent
Dependent
Substitution Method
In a system to two equations with two
variables, the substitution method involves
using one equation to find an expression for
one variable in terms of the other, then
substituting into the other equation of the
system.
Solve the system.
3x  2 y  11
x  y3
Elimination Method
Another way to solve a system of two
equations, called the elimination method,
uses multiplication and addition to eliminate
a variable from one equation. To eliminate a
variable, the coefficients of that variable in
the two equations must be additive inverses.
Elimination Method
To achieve this, we use properties of algebra
to change the system to an equivalent
system, one with the same solution set. The
three transformations produce an equivalent
system are listed here.
Elimination Method
Solve the system.
3x  4 y  1
2 x  3 y  12
Solve the system.
3x  2 y  4
 6x  4 y  7
Solve the system.
8x  2 y  - 4
 4x  y  2
Applying Systems of Equations
Many applied problems involve more than
one unknown quantity. Although some
problems with two unknowns can be solved
using just one variable, it is often easier to
use two variables
Applying Systems of Equations
To solve a problem with two unknowns, we
must write two equations that relate the
unknown quantities. The system formed by
the pair of equations can then be solved
using the methods of this chapter.
Title IX legislation prohibits sex
discrimination in sports programs. In 1997
the national average spent on two varsity
athletes, one female and one male, was
$6050 for Division I-A schools. However,
average expenditures for a male athlete
exceeded those for a female athlete by
$3900. Determine how much was spent per
varsity athlete for each gender.
x is the male expenditures.
y is the female expenditures.
In 1997 the national average spent on two varsity
athletes, one female and one male, was $6050 for
Division I-A schools.
x y
 6050
2
However, average expenditures for a male athlete
exceeded those for a female athlete by $3900.
x  6050  y
x y
 6050
2
x  6050  y

x  y  12100

x  y  3900
x  y  12100 8000 - y  3900
x  y  3900
2x
 16000 8000 - 3900  y
4100  y
x
 8000
Solving Linear Systems with Three Unknowns
(Variables)
Earlier we saw that the graph of a linear
equation in two unknowns is a straight line.
The graph of a linear equation in three
unknowns requires a three-dimensional
coordinate system.
Solving Linear Systems with Three Unknowns
(Variables)
The three number lines are placed at right
angles. The graph of a linear equation in
three unknowns is a plane.
Some possible intersections of planes
representing three equations in three variables are
shown.
To solve a linear system with three
unknowns, first eliminate a variable from
any two of the equations. then eliminate the
same variable from a different pair of
equations.
Eliminate a second variable using the
resulting two equations in two variables to
get an equation with just one variable whose
vlaue you can now determine.
Find the values of the remaining variables
by substitution. Solutions of the system are
written as ordered pairs.
Solve the system.
3x  9 y  6 z  3
2x  y  z  2
x yz 2
Solve the system.
x  2y  z  4
3 x  y  4 z  9
Using Systems of Equations to Model Data
Applications with three unknowns usually
require solving a system of three equations.
We can find the equation of a parabloa in
the form
y = ax2 + bx + c
by solving a system of three equations with
three variables.
Find the equation of the parabola
y = ax2 + bx + c
that passes through the points (2,4), (-1,1), and (-2,5)
4  a 2   b2   c
or
4  4a  2b  c
1  a - 1  b- 1  c
or
1 a b c
1  a - 2   b- 2   c
or
5  4a  2b  c
2
2
2
An animal feed is made from three
ingredients: corn, soybeans, and cottonseed.
On unit of each ingredient provides units of
protein, fat, and fiber as shown in the table.
How many units of each ingredient should
be used to make a feed that contains 22 units
of protein, 28 units of fat, and 18 units of
fiber.
.25x  .4y  .2z  22
.4x  .2y  .3z  28
.3x  .2y  .1z  18