Math 90
3.3 "Solving Systems of Equations by Graphing"
Objectives:
*
Determine whether an ordered pair is a solution of a system of equations.
*
Solve systems of two linear equations in two variables by graphing.
Systems of Equations and Solutions
Solution of a System of Equations:
kA solution of a system of two equations is an ordered pair that makes both equations true.k
Example 1: (Checking for solutions for systems of equations)
Determine whether
( the given ordered pair is a solution of the system of equations.
(
x = 2y + 8
3y 2x = 2
a) (2; 3) ;
b) (4; 2) ;
2x + y = 1
y + 2x = 8
Graphing Systems of Equations
Recall that the graph of an equation is a drawing that represents its solution set. If the graph of an equation is a line,
then every point on the line corresponds to an ordered pair that is a solution of the equation. If we graph a system of two
linear equations, we graph both equations and …nd the coordinates of the points of intersection, if any exist.
Example 2: (Solving systems of equations by graphing)
Solve
( the systems of equations by graphing.
x+y =3
a)
3x y = 1
y
b)
(
4x = 3 (4
y)
2y = 4 (3
x)
y
4
2
6
4
2
-4
-2
2
4
x
-2
-4
-2
2
4
x
-2
-4
Page: 1
Notes by Bibiana Lopez
Introductory Algebra by Marvin L. Bittinger
c)
(
y=
x
3.3
x
d)
y=0
(
x=3
y=
2
y
-4
y
4
4
2
2
-2
2
4
-4
x
-2
-2
2
4
-2
-4
x
-4
Inconsistent Systems
Sometimes a system of equations will have no solution. These systems are called inconsistent systems.
Example 3: (Solving systems of equations by graphing)
Solve
( the systems of equations by graphing.
2x + y = 6
a)
4x + 2y = 8
b)
(
y
3x
6y = 18
x = 2y + 3
y
4
4
2
-4
-2
2
2
-2
4
x
-4
-2
2
4
6
x
-2
-4
-6
-4
Page: 2
Notes by Bibiana Lopez
Introductory Algebra by Marvin L. Bittinger
3.3
Dependent Equations
Sometimes a system will have in…nitely many solutions. In this case, we say that the equations of the system are
dependent equations.
Example 4: (Solving systems of equations by graphing)
Solve
( each system of equations by graphing.
y 2x = 4
a)
4x + 8 = 2y
(
b)
2x = 3 (2
y)
3y = 2 (3
x)
y
-4
y
4
4
2
2
-2
2
4
-2
-4
x
-2
2
4
x
-2
-4
-4
When we graph a system of two equations in two variables, we obtain one of the following three results.
4
y = 2x + 3 2
-4
-2
4
2
2y = 4x + 6
2
4
-2
-4
-4
2y = -x + 4
-2
y=x+2
4
y = 3x + 2 2
2
4
-2
-4
-4
-2
2
4
-2
-4
y = 3x - 1
In…nitely many solutions: Same Graph.
One solution: Graphs intersect
No solution: Graphs are parallel.
Consistent System
Consistent System
Inconsistent System
Page: 3
Notes by Bibiana Lopez