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TMAT 103
Chapter 6
Systems of Linear Equations
TMAT 103
§6.1
Solving a System of Two Linear
Equations
§6.1 – Solving a System of Two
Linear Equations
• Systems of two linear equations:
a1 x  b1 y  c1
a2 x  b2 y  c2
§6.1 – Solving a System of Two
Linear Equations
•
Graphs of linear systems of equations with two
variables
1. The two lines may intersect at a common, single
point. This point, in ordered pair form(x, y), is the
solution of the system
•
independent and consistent
2. The two lines may be parallel with no points in
common; hence, the system has no solution
•
inconsistent
3. The two lines may coincide; the solution of the
system is the set of all points on the common line
•
dependent
§6.1 – Solving a System of Two
Linear Equations
•
Methods available to solve systems of
equations
1. Addition-subtraction method
2. Method of substitution
§6.1 – Solving a System of Two
Linear Equations
•
Solving a pair of linear equations by the
addition-subtraction method
1.
2.
3.
4.
5.
If necessary, multiply each side of one or both equations
by some number so that the numerical coefficients of one
of the variables are of equal absolute value.
If these coefficients of equal absolute value have like
signs, subtract one equation from the other. If they have
unlike signs, add the equations.
Solve the resulting equation for the remaining variable.
Substitute the solution for the variable found in step 3 in
either of the original equations, and solve this equation for
the second variable.
Check
§6.1 – Solving a System of Two
Linear Equations
•
Examples – solve the following using the
addition-subtraction method
5 x  3 y  16
4 x  2 y  18
2 x  3 y  2
4x  6 y 1
§6.1 – Solving a System of Two
Linear Equations
•
Solving a pair of linear equations by the method
of substitution
1.
2.
3.
4.
5.
From either of the two given equations, solve for one
variable in terms of the other.
Substitute this result from step 1 in the other equation.
Note that this step eliminates one variable.
Solve the equation obtained from step 2 for the remaining
variable.
From the equation obtained in step 1, substitute the
solution for the variable found in step 3, and solve this
resulting equation for the second variable.
Check
§6.1 – Solving a System of Two
Linear Equations
•
Examples – solve the following using the
method of substitution
2x  y  7
 9 x  y  13
2 x  12 y  2
5 x  54 y  5
§6.1 – Solving a System of Two
Linear Equations
•
Steps for problem solving
1.
2.
3.
4.
5.
6.
7.
8.
Read the problem carefully at least two times.
If possible, draw a picture or diagram.
Write what facts are given and what unknown quantities
are to be found.
Choose a symbol to represent each quantity to be found.
Write appropriate equations relating these variables from
the information given in the problem (there should be one
equation for each unknown).
Solve for the unknown variables using an appropriate
method.
Check your solution in the original equation.
Check your solution in the original verbal problem.
§6.1 – Solving a System of Two
Linear Equations
•
Examples – solve the following
A plane can travel 900 mile with the wind
in 3 hours. It makes the return trip in 3.5
hours. Find the rate of windspeed, and the
speed of the plane
A chemist has a 5% solution and an 11%
solution of acid. How much of each must
be mixed to get 1000L of a 7% solution?
TMAT 103
§6.2
Other systems of equations
§6.2 – Other systems of equations
•
Other types of problems can be solved
using either the addition-subtraction
method, or the method of substitution
–
Literal equations
•
•
–
coefficients are letters
will not be covered in this class
Non-linear equations
•
variables in denominator
§6.2 – Other systems of equations
•
Examples – solve the following using the
method of substitution or additionsubtraction method
3
x
 12y  32
4
x
 9y  21 23
TMAT 103
§6.3
Solving a System of Three Linear
Equations
§6.3 – Solving a System of Three
Linear Equations
• Systems of three linear equations:
a1 x  b1 y  c1 z  d1
a2 x  b2 y  c2 z  d 2
a3 x  b3 y  c3 z  d 3
§6.3 – Solving a System of Three
Linear Equations
•
Graphs of linear systems of equations with three
variables
1.
2.
3.
4.
The three planes may intersect at a common, single point.
This point, in ordered triple form (x, y, z), is then the
solution of the system.
The three planes may intersect along a common line. The
infinite set of points that satisfy the equation of the line is
the solution of the system.
The three planes may not have any points in common; the
system has no solution. For example, the planes may be
parallel, or they may intersect triangularly with no points
in common to all three planes.
The three planes may coincide; the solution of the system
is the set of all points in the common plane.
§6.3 – Solving a System of Three
Linear Equations
•
Solving a pair of linear equations by the
addition-subtraction method
1. Eliminate a variable from any pair of equations using
the same technique from section 6.1
2. Eliminate the same variable from any other pair of
equations.
3. The results of steps 1 and 2 is a pair of linear
equations in two unknowns. Solve this pair for the
two variables
4. Solve for the third variable by substituting the results
from step 3 in any one of the original equations
5. Check
§6.3 – Solving a System of Three
Linear Equations
•
Examples – solve the following using the
addition-subtraction method
x  3 y  6 z  1
2 x  y  z  10
5 x  2 y  3 z  27
§6.3 – Solving a System of Three
Linear Equations
•
Example – solve the following
75 acres of land were purchased for
$142,500. The land facing the highway
cost $2700/acre. The land facing the
railroad cost $2200/acre, and the
remainder cost $1450/acre. There were 5
acres more facing the railroad than the
highway. How much land was sold at
each price?