Lecture 19 section 8.1 system of equns

MATH 107
Section 8.1
Systems of Linear Equations;
Substitution and Elimination

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Definitions
A set of equations with common variables is called a
system of equations.
If each equation is linear, then it is a system of linear
equations or a linear system of equations.
If at least one equation is nonlinear, then it is called
a nonlinear system of equations.
Here’s a system of two linear equations in two
variables

Definitions
A system of equations is sometimes referred to as a
set of simultaneous equations.
A solution of a system of equations in two variables
x and y is an ordered pair of numbers (a, b) such that
when x is replaced by a and y is replaced by b, all
resulting equations in the system are true.
The solution set of a system of equations is the set
of all solutions of the system.

EXAMPLE 1 Verifying a Solution
Verify that the ordered pair (3, 1) is the solution
(3, 1) satisfies both equations, so it is the solution.
of the system of linear equations
Solution
Replace x with 3 and y with 1.

EXAMPLE 2 Solving a System by the Graphical Method
Use the graphical method to solve the system of
equations
Solution
Step 1 Graph both equations on the same
coordinate axes.

Solution continued
x-intercept is 6; y-intercept is 4
(ii) Find intercepts of equation (2).
(i) Find intercepts of equation (1).
a. Set x = 0 in 2x – y = 4 and solve for y:
2(0) – y = 4, or y = –4
so the y-intercept is –4.
b. Set y = 0 in 2x – y = 4 and solve for x:
2x – 0 = 4, or x = 2
so the x-intercept is 2.

Solution continued
Step 2 Find the
point(s) of
intersection of
the two graphs.
The point of
intersection of the two
graphs is (3, 2).

Solution continued
Step 3 Check your solution(s).
The solution set is {(3, 2)}.
Replace x with 3 and y with 2.
Step 4 Write the solution set for the system.

SOLUTIONS OF SYSTEMS OF
EQUATIONS
The solution set of a system of two linear
equations in two variables can be classified in
one of the following ways.
1. One solution.
The system is
consistent and
the equations are
said to be
independent.

EQUATIONS
2. No solution. The
lines are parallel.
The system is
inconsistent.

EQUATIONS
3. Infinitely many
solutions. The
lines coincide.
The system is
consistent and
the equations are
said to be
dependent.

OBJECTIVE Reduce the solution of the
system to the solution of one equation in
one variable by substitution.
Step 1 Choose one of the equations and
express one of its variables in terms of the
other variable.
EXAMPLE 3 The Substitution Method
EXAMPLE Solve the system.
1. In equation (2), express y in terms of x.
y = 2x + 9

Step 2 Substitute the expression found in
Step 1 into the other equation to obtain an
equation in one variable.
Step 3 Solve the equation obtained in Step
2.
2.
3.

Step 4 Substitute the value(s) you found in
Step 3 back into the expression you found
in Step 1. The result is the solution(s).
4.
The solution set is {(−6, −3)}.

Step 5 Check your answer(s) in the
original equations.
5. Check: x = −6 and y = −3

Lecture 19 section 8.1 system of equns

OBJECTIVE Solve a system of two linear
equations by first eliminating one variable.
Step 1 Adjust the coefficients. If
necessary, multiply both equations by
appropriate numbers to get two new
equations in which the coefficients of the
variable to be eliminated are opposites.
EXAMPLE 6 The Elimination Method
1. Select y as the variable to be eliminated.
Multiply equation (1) by 4 and equation (2)
by 3.

Step 2 Add the resulting equations to get
an equation in one variable.
2.

Step 3 Solve the resulting equation.
3.

Step 4 Back-substitute the value you found
into one of the original equations to solve
for the other variable.
4.

Step 5 Write the solution set from Steps 3
and 4.
Step 6 Check your solution(s) in the
original equations (1) and (2).
5. The solution set is {(9, 1)}.
6. Check x = 9 and y = 1.

EXAMPLE 4 Attempting to Solve an Inconsistent System of Equations
Solve the system of equations.
Step 1 Solve equation (1) for y in terms of x.
Solution
Step 2 Substitute into equation (2).

EXAMPLE 4
Since the equation 0 = 3
is false, the system is
inconsistent. The lines
are parallel, do not
intersect and the system
has no solution.
Solution continued
Attempting to Solve an Inconsistent System of Equations
Step 3 Solve for x.

EXAMPLE 5 Solving a Dependent System
Solve the system of equations.
Step 1 Solve equation (2) for y in terms of x.
Solution

Solution continued
Step 2 Substitute (6 – 2x) for y in equation (1).
The equation 0 = 0 is true for every value of x.
Thus, any value of x can be used in the equation
y = 6 – 2x for back substitution.
Step 3 Solve for x.

Solution continued
The solutions are of the form (x, 6 – 2x) and
the solution set is {(x, 6 – 2x)}.
The solution set
consists of all ordered
pairs (x, y) lying on the
line with equation 4x +
2y = 12. The system
has infinitely many
solutions.

Lecture 19 section 8.1 system of equns

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Lecture 19 section 8.1 system of equns