solving equations with rational expressions distinguish between operations with rational expressions...
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Solving Equations with Rational Expressions
Distinguish between operations with rational expressions and equations with terms that are rational expressions.
Solve equations with rational expressions.
Solve a formula for a specified variable.
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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.
Solution:
Identify each of the following as an expression or an equation. Then simplify the expression or solve the equation.
5
2 3 6
x x
2 4
3 9
x x
equation expression
6 65
2 3 6
x x 3
3
3
2 4
9
x x
3 2 5x x 5x
2
9
x
6 4
9 9
x x
5
EXAMPLE 1 Distinguishing between Expressions and Equations
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.
Solve, and check the solution.
Solution:
2 3 6
5 3 5
m m
2 3 6
515 1
55
3
m m
6 9 5 18m m 9 189 9m
9m
Check:
1
2 3 6
5 35
91
55
9
2 3 6
5 3 5
m m
3 21 5 3 18
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.
EXAMPLE 2 Solving an Equation with Rational Expressions
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.
Solve equations with rational expressions. (cont’d)
Solution:
2 21
1 1
x
x x
Solve, and check the proposed solution.
1
1 12 2
11
x
xx
xx
1 2 2x xx 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
Solution:
2 2
2 3
2p p p p
Solve, and check the proposed solution.
2 66 6p
22 22 3 6pp pp
4p
2 3
22 1 2
11p p p
pp p p
p p p
The solution set is {4}.
EXAMPLE 3 Solving an Equation with Rational Expressions
Solution:
2
8 3 3
4 1 2 1 2 1
r
r r r
Solve, and check the proposed solution.
2 1 2 1 2 18 3 3
2 1 2 1 2 1 2 12 1r r r
r
r r rr
r
8 6 3 6 3r r r
8 1212 12r rr r 4 0r
0r
Since 0 does not make any denominators equal 0, the solution set is {0}.
EXAMPLE 5 Solving an Equation with Rational Expressions
Solve, and check the proposed solution (s).
2
1 1 2
2 5 5 4x x
1 1 2
2 5 55 2 2 5
2 22 2x x x
x x xx
25 10 4 222x x 2 5 4 0x x
1 4 0x x
1 11 0x
4x 1x
Solution:
The solution set is {−4, −1}.
4 44 0x or
EXAMPLE 6 Solving an Equation with Rational Expressions
2
6 1 4
5 10 5 3 10x x x x
Solve, and check the proposed solution.
Solution:
6 1 4
5 2 55 2 5 5 2
2 55
x xx x
x xx x
6 30 5 10 20x x
40 40 04 20x
60x
The solution set is {60}.
EXAMPLE 7 Solving an Equation with Rational Expressions
for x
z yx y
Solve each formula for the specified variable.
Solution: xz
xx
yx y y
z x y x
z z
xy
zx xx
xy x
z
for s t
b sr
br s t
( )1
s rr
tb
r
b xt tt
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.
br t s
EXAMPLE 8 Solving for a Specified Variable
Solve the following formula for z.
Solution:
2 1 1
x yxy
zz xyz
2yz x
z z
z xy
2xy
y x xz
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.
1 2y x
z xy
2
xyz
y x
EXAMPLE 9 Solving for a Specified Variable
21 1xy
y xzxy xy
2 1 1
x y z