7.1 Systems of Linear Equations: Two Equations Containing Two Variables

1
7.1Systems of Linear Equations Two Equations
Containing Two Variables
2
A system of equations is a collection of two or
more equations, each containing one or more
variables.
A solution of a system of equations consists of
values for the variables that reduce each
equation of the system to a true statement.
To solve a system of equations means to find all
solutions of the system.
When a system of equations has at least one
solution, it is said to be consistent otherwise
it is called inconsistent.
3
An equation in n variables is said to be linear
if it is equivalent to an equation of the form
where
are n distinct variables,
are constants, and at least one of the
as is not zero.
4
If each equation in a system of equations is
linear, then we have a system of linear equations.
5
If the graph of the lines in a system of two
linear equations in two variables intersect, then
the system of equations has one solution, given
by the point of intersection. The system is
consistent and the equations are independent.
y
Solution
x
6
If the graph of the lines in a system of two
linear equations in two variables are parallel,
then the system of equations has no solution,
because the lines never intersect. The system is
inconsistent.
y
x
7
If the graph of the lines in a system of two
linear equations in two variables are coincident,
then the system of equations has infinitely many
solutions, represented by the totality of points
on the line. The system is consistent and
dependent.
y
x
8
Two Algebraic Methods for Solving a System
1. Method of substitution 2. Method of
elimination
9
Use Method of Substitution to solve
(1)
(2)
STEP 1 Solve for x in (2)
add x and subtract 2 on both sides
10
STEP 2 Substitute for x in (1)
STEP 3 Solve for y
11
STEP 4 Substitute y -11/5 into result in
Step1.
Solution
12
STEP 5 Verify solution
in
13
Rules for Obtaining an Equivalent System of
Equations
1. Interchange any two equations of the system.
2. Multiply (or divide) each side of an equation
by the same nonzero constant.
3. Replace any equation in the system by the sum
(or difference) of that equation and any other
equation in the system.
14
(1)
Use Method of Elimination to solve
(2)
Multiply (2) by 2
Replace (2) by the sum of (1) and (2)
Equation (2) has no solution. System is
inconsistent.
15
(1)
Use Method of Elimination to solve
(2)
Multiply (2) by 2
Replace (2) by the sum of (1) and (2)
The original system is equivalent to a system
containing one equation. The equations are
dependent.
16
This means any values x and y, for which 2x -y 4
represent a solution of the system.
Thus there is infinitely many solutions and they
can be written as
or equivalently