Transcript Algebra 2 - PowerPoint Notes

7.6 Solving Equations with Rational Expressions
1
Distinguish between operations with rational expressions and
equations with terms that are rational expressions.
2
Solve equations with rational expressions.
3
Solve a formula for a specified variable.
Distinguish between operations with rational
expressions and equations with terms that are
rational expressions.
Before solving equations with rational expressions, you must
understand the difference between sums and differences of terms with
rational coefficients, or rational expressions, and equations with terms
that are rational expressions.
Sums and differences are expressions to simplify. Equations are
solved.
Uses of the LCD
When adding or subtracting rational expressions, keep the LCD
throughout the simplification.
When solving an equation, multiply each side by the LCD so the
denominators are eliminated.
EXAMPLE 1 Distinguishing between Expressions and Equations
Identify each of the following as an expression or an equation. Then
simplify the expression or solve the equation.
x x 5
 
2 3 6
Solution:
equation
 x x 5
6     6
2 3 6
3x  2x  5
x5
5
2x 4x

3
9
expression
3 2x 4x
  
3 3
9
6x 4x


9
9
2x

9
Solve equations with rational expressions.
When an equation involves fractions, we use the multiplication
property of equality to clear the fractions. Choose as multiplier the
LCD of all denominators in the fractions of the equation.
Recall that the denominator of a rational expression cannot equal 0,
since division by 0 is undefined. Therefore, when solving an
equation with rational expressions that have variables in the
denominator, the solution cannot be a number that makes the
denominator equal 0.
EXAMPLE 2 Solving an Equation with Rational Expressions
Solve, and check the solution.
2m  3 m
6
 
5
3
5
Check:
2m  3 m
6
 
5
3
5
 2m  3 m   6 
15 
     15
 2  9   3  9  
6
3   5  15  
 5

   15
5
3 
5

6m  9  5m  18
3  21  5  3  18
Solution:
m  9  9  18  9
m  9
63  45  18
18  18
The use of the LCD here is different from its use in simplifying rational
expressions. Here, we use the multiplication property of equality to multiply
each side of an equation by the LCD. Earlier, we used the fundamental
property to multiply a fraction by another fraction that had the LCD as both its
numerator and denominator.
Solve equations with rational expressions. (cont’d)
While it is always a good idea to check solutions to guard against
arithmetic and algebraic errors, it is essential to check proposed
solutions when variables appear in denominators in the original
equation.
Solving an Equation with Rational Expressions
Step 1: Multiply each side of the equation by the LCD to clear the
equation of fractions. Be sure to distribute to every term on
both sides.
Step 2: Solve the resulting equation.
Step 3: Check each proposed solution by substituting it into the
original equation. Reject any that cause a denominator to
equal 0.
EXAMPLE 3 Solving an Equation with Rational Expressions
Solve, and check the proposed solution.
2
2x
1

x 1 x 1
Solution:
2  2x

 x  1 1 
 x  1

 x 1  x 1
 x  1  2  x  2x  x
1  x
When the equatioin is solved, − 1 is a proposed solution.
However, since x = − 1 leads to a 0 denominator in the original
equation, the solution set is Ø.
EXAMPLE 3 Solving an Equation with Rational Expressions
Solve, and check the proposed solution.
2
3
 2
2
p 2p p  p
Solution:

 

2
3
p  p  2  p  1 
  
 p  p  2  p  1
 p  p  2    p  p  1 
2p  2  2p  3p  6  2p
2  6  p  6  6
p4
The solution set is {4}.
EXAMPLE 5 Solving an Equation with Rational Expressions
Solve, and check the proposed solution.
8r
3
3


2
4r  1 2r  1 2r  1
Solution:

  3
8r
3 

 2r  1 2r  1 
  
  2r  1 2r  1
  2r  1 2r  1   2r  1 2r  1 
8r  6r  3  6r  3
8r 12r  12r 12r
4r  0
r 0
Since 0 does not make any denominators equal 0, the solution
set is {0}.
EXAMPLE 6 Solving an Equation with Rational Expressions
Solve, and check the proposed solution (s).
1
1
2
 
x  2 5 5  x2  4
Solution:
 1

1 
2
5  x  2  x  2  
   
 5  x  2  x  2 
  x  2  5   5  x  2  x  2  
5 x  10  x 2  4  2  2  2
x2  5x  4  0
 x  1 x  4  0
x  1 1  0 1
x  1
The solution set is {−4, −1}.
or
x44  04
x  4
EXAMPLE 7 Solving an Equation with Rational Expressions
Solve, and check the proposed solution.
6
1
4

 2
5 x  10 x  5 x  3x  10
Solution:


6
1  
4
5  x  2  x  5  

  
 5  x  2  x  5 
 5  x  2   x  5     x  2  x  5  
6 x  30   5x  10  20
x  40  40  20  40
x  60
The solution set is {60}.
EXAMPLE 8 Solving for a Specified Variable
Solve each formula for the specified variable.
st
b
for s
r
Solution:
st  r 
b( r ) 
 
r 1
br  s  t
b t  x t t
br  t  s
x
z
for y
x y
x
z  x  y 
 x  y
x y
z  x  y x

z
z
x yx 
x
x
z
x
y x
z
Remember to treat the variable for which you are solving as if it were the
only variable, and all others as if they were contants.
EXAMPLE 9 Solving for a Specified Variable
Solve the following formula for z.
2 1 1
 
x y z
Solution:
2 1 1
xyz       xyz
 x  y z
2 yz xz  xy

z
z
xy
2y  x  x   x
z
 1 
 xy   1 
   2 y  x     
 z   xy 
 xy 
1 2y  x

z
xy
xy
z
2y  x
When solving an equation for a specified variable, be sure that the
specified variable appears alone on only one side of the equals symbol in
the final equation.