Download 7.1 Systems of Linear Equations: Two Equations Containing Two

7.1
Systems of Linear Equations:
Two Equations Containing Two
Variables
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.
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 a’s is not zero.
If each equation in a system of equations
is linear, then we have a system of linear
equations.
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
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
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
Two Algebraic Methods for Solving
a System
1. Method of substitution
2. Method of elimination
Use Method of Substitution to solve:
(1)
(2)
STEP 1: Solve for x in (2)
add x and subtract 2
on both sides
STEP 2: Substitute for x in (1)
STEP 3: Solve for y
STEP 4: Substitute y = -11/5 into result in
Step1.
Solution:
STEP 5: Verify solution
in
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.
Use Method of Elimination
to solve:
Multiply (2) by 2
Replace (2) by the sum of (1) and (2)
Equation (2) has no solution. System is
inconsistent.
(1)
(2)
Use Method of Elimination
to solve:
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.
(1)
(2)
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