Page 452 - Algebra 1

Example
2
Solving Dependent Systems of Equations
Solve.
x + 3y = 6
_1 x + y = 2 3
SOLUTION Write the equations in slope-intercept form. Method1: x+3y=6 y=-_1x+2
3
_1 x + y = 2 y = - _1 x + 2 33
The equations are identical. Since the graphs would be the same line, there
are infinitely many solutions. Any ordered pair (x, y) that satifies the
e q u a t i o n y = - _1 x + 2 . 3
SOLUTION
Method2: x+3y=6 _1 x + y = 2
3
Math Reasoning
Analyze What is characteristic of systems of equations that are dependent?
y = 2 -_1 x 3
Isolate y in the second equation.
S u b s t i t u t e 2 - _1 x f o r y i n t h e f i r s t e q u a t i o n .
x + 3 ( 2 - _1 x ) = 6 33
x+6-x =6 Distribute. 6 = 6 Simplify.
All variables have been eliminated. The last equation is true, 6 = 6. This means that the original equations are true for all values of the variables. There are infinitely many solutions—an infinite set of ordered pairs.
A consistent system will have at least one common solution. An independent system will have exactly one solution. An independent system is also a consistent system. The graphs of the equations of an independent system will intersect at one point. Systems of linear equations can be classified into three different categories based on the number of solutions.
Systems of Linear Equations
Consistent and Independent
Consistent and Dependent
Inconsistent
Exactly One Solution
Infinitely Many Solutions
No Solution
y
x
The graphed lines intersect at a single point.
y
x
The graphed lines are the same line. The line is the solution.
y
x
The lines are parallel and do not intersect.
Lesson 67 437