Page 74 - Elementary Algebra Exercise Book I

ELEMENTARY ALGEBRA EXERCISE BOOK I equAtions

Solution: The system is equivalent to

lg |x + y| = lg 10,
y
lg = lg 2,
|x|

which lead to

|x + y| = 10,
y =2|x|.

y> 0 is always true since y =2|x| and x =0.

When x> 0, the system becomes

x + y = 10,
y =2x,

whose solution is x = 10/3,y = 20/3.

When x< 0, the system become

x + y = 10,
y = −2x,

whose solution is x = −10,y = 20.

We can verify that (10/3, 20/3), (−10, 20) indeed are solutions of the original system.

√ √ √
2.75 Solve the equation 2x + x + x +2+2 x +2x − 4= 0.
2
√
√
√ √
√
√
√
√
Solution: The equation is equivalent to x+2 x x + 2+x+2+ x+ x +2−6 =0 ⇔ ( x+ x + 2) +( x+ x + 2)−6=
2
√ √
√
√
√
√
√
√
x+2 x x + 2+x+2+ x+ x +2−6 =0 ⇔ ( x+ x + 2) +( x+ x + 2)−6= 0.
2
0 √ √
Let y = x + x +2 ( y> 0), then y + y − 6= 0 ⇒ (y − 2)(y + 3) = 0 ⇒ y =2 or
2
y = −3 (deleted). Hence, √ x + √ x +2 = 2 ⇒ √ x +2 = 2 − √ x ⇒ x +2 = 4 − 4 x + x ⇒ √ x =1/2 ⇒
√
√ √ √ √ √ √
x + x +2 = 2 ⇒ x +2 = 2 − x ⇒ x +2 = 4 − 4 x + x ⇒ x =1/2 ⇒ x =1/4 ,
x =1/4
which is the root of the original equation.
2.76 Solve the system of equations
x + y + z =3,
2
2
x + y + z 2 =3,
5
5
x + y + z 5 =3.
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