Ejercicio 144

CAPITULO XVI

Ecuaciones literales de primer grado con una icognita
Ejercicio 144
Resolver las siguientes ecuaciones:
  1. m x 1 m = 2 m m 2 x m x = 2 m m 2 x =2x m 2 =2x+x m 2 =3x x = m 2 3
  2. a x + b 2 = 4a x 2a+bx 2 x = 4a x 2a+bx =8a bx =8a2a bx =6a x = 6a b
  3. x 2a 1x a 2 = 1 2a ax1+x 2 a 2 = 1 2a ax1+x =a x (a+1 ) = a+1 x =1
  4. m x + n m = n x +1 m 2 +nx m x = n+x x m 2 +nx =m(n+x ) m 2 +nx =mn+mx m 2 mn =mxnx m (mn ) =x (mn ) x =m
  5. a1 a + 1 2 = 3a2 x 2(a1 ) +a 2a = 3a2 x 2a2+a 2a = 3a2 x 3a2 2a = 3a2 x x (3a2 ) =2a (3a2 ) x =2a
  6. ax a bx b = 2(ab ) ab b(ax ) a(bx ) ab = 2(ab ) ab ab bx ab +ax =2(ab ) x (ab ) =2 (ab ) x =2
  7. x3a a 2 2ax ab = 1 a b(x3a ) a(2ax ) a 2 b = 1 a bx3ab2 a 2 +ax =ab ax+bx =2 a 2 +3abab ax+bx =2 a 2 +2ab x (a+b ) =2a (a+b )
  8. x+m m x+n n = m 2 + n 2 mn 2 n(x+m ) m(x+n ) mn = m 2 + n 2 2mn mn nx+ mn mx mn = n 2 2mn+ m 2 x (nm ) = (nm ) 2 x =nm
  9. xb a =2 xa b xb a + xa b =2 b(xb ) +a(xa ) ab =2 bx b 2 +ax a 2 =2ab ax+bx = a 2 +2ab+ b 2 x (a+b ) = (a+b ) 2 x =a+b
  10. 4x 2a+b 3 = 3 2 4x 2a+b =3 3 2 4x 2a+b = 63 2 4x 2a+b = 3 2 2(4x ) =3(2a+b ) 8x =3(2a+b ) x = 3(2a+b ) 8
  11. 2a+3x x+a = 2(6xa ) 4x+a 2a+3x x+a = 12x2a 4x+a (2a+3x ) (4x+a ) =(12x2a ) (x+a ) 8ax+2 a 2 + 12 x 2 +3ax = 12 x 2 +12ax2ax2 a 2 2 a 2 +2 a 2 =10ax11ax 4 a 2 = a x x =4a
  12. 2(xc ) 4xb = 2x+c 4(xb ) 2x2c 4xb = 2x+c 4x4b (2x2c ) (4x4b ) =(2x+c ) (4xb ) 8 x 2 8bx8cx+8bc = 8 x 2 2bx+4cxbc 8bx8cx4cx+2bx =8bcbc 6bx12cx =9bc x(b+c ) =bc x = 3bc 2(b+c )
  13. 1 n m x = 1 mn 1 x xmn nx = xmn m nx m(xmn ) =xmn mx m 2 n =xmn mxx = m 2 nmn x (m1 ) =mn (m1 ) x =mn
  14. (x2b ) (2x+a ) (xa ) (a2b+x ) =2 (x2b ) (2x+a ) =2(xa ) (a2b+x ) 2 x 2 +ax4bx2ab =2( ax 2bx+ x 2 a 2 +2ab ax ) 2 x 2 +ax 4bx 2ab = 4bx + 2 x 2 2 a 2 +4ab ax =2ab2 a 2 +4ab ax =6ab2 a 2 a x =2 a (3ba ) x =2(3ba )
  15. x+m xn = n+x m+x (x+m ) (m+x ) =(n+x ) (xn ) (x+m ) 2 = x 2 n 2 x 2 +2mx+ m 2 = x 2 n 2 2mx = m 2 n 2 x = m 2 + n 2 2m
  16. x(2x+3b ) (x+b ) x+3b =2 x 2 bx+ b 2 x(2 x 2 +3bx+2bx+3 b 2 ) =(x+3b ) (2 x 2 bx+ b 2 ) 2 x 3 + 5b x 2 +3 b 2 x = 2 x 3 b x 2 + b 2 x+ 6b x 2 3 b 2 x+3 b 3 3 b 2 x+2 b 2 x =3 b 3 5 b 2 x =3 b 3 x = 3b 5
  17. 3 4 ( x b + x a ) = 1 3 ( x b x a ) + 5a+13b 12a 3 4 ( ax+bx ab ) = 1 3 [ x b x a + 5a+13b 4a ] 3 4 ( ax+bx ab ) = 1 3 [ 4ax4bx+5ab+13 b 2 4 ab ] 9(ax+bx ) =4ax4bx+5ab+13 b 2 9ax+9bx4ax+4bx =5ab+13 b 2 5ax+13bx =b(5a+13b ) x (5a+13b ) =b (5a+13b ) x =b
  18. x+a 3 = (xb ) 2 3xa + 3ab3 b 2 9x3a x+a 3 = (xb ) 2 3xa + 3 (ab b 2 ) 3 (3xa ) x+a 3 = 1 3xa [ (xb ) 2 +ab b 2 ] x+a 3 = 1 3xa [ x 2 2bx+ b 2 +ab b 2 ] (x+a ) (3xa ) =3( x 2 2bx+ab ) 3 x 2 ax+3ax a 2 = 3 x 2 6bx+3ab 2ax+6bx = a 2 +3ab 2x (a+3b ) =a (a+3b ) x = a 2
  19. 5x+a 3x+b = 5xb 3xa (5x+a ) (3xa ) =(5xb ) (3x+b ) 15 x 2 5ax+3ax a 2 = 15 x 2 +5bx3bx b 2 5ax+3ax5bx+3bx = a 2 b 2 2ax2bx =(ab ) (a+b ) 2x (a+b ) =(ab )(a+b ) x = ba 2
  20. x+a xa xa x+a = a(2x+ab ) x 2 a 2 (x+a ) 2 (xa ) 2 (xa ) (x+a ) = a(2x+ab ) x 2 a 2 [(x+a ) +(xa ) ] [(x+a ) (xa ) ] =2ax+ a 2 b (x+ a +x a ) ( x +a x +a ) =2ax+ a 2 b 2x(2a ) =2ax+ a 2 b 4ax2ax = a 2 b 2 a x = a 2 b x = ab 2
  21. 2x3a x+4a 2 = 11a x 2 16 a 2 2x3a2(x+4a ) x+4a = 11a (x+4a )(x4a ) 2x 3a 2x 8a = 11a x4a 11a (x4a ) = 11a x+4a =1 x =4a1
  22. 1 x+a + x 2 a 2 +ax = x+a a 1 x+a + x 2 a(x+a ) = x+a a 1 x+a (1+ x 2 a ) = x+a a 1 x+a ( a+ x 2 a ) = x+a a a+ x 2 = (x+a ) 2 a+ x 2 = x 2 +2ax+ a 2 a a 2 =2ax a (1a ) =2 a x x = 1a 2
  23. 2(a+x ) b 3(b+x ) a = 6( a 2 2 b 2 ) ab 2a(a+x ) 3b(b+x ) ab = 6( a 2 2 b 2 ) ab 2 a 2 +2ax3 b 2 3bx =6 a 2 12 b 2 2ax3bx =6 a 2 2 a 2 12 b 2 +3 b 2 x(2a3b ) =4 a 2 9 b 2 x (2a3b ) =(2a+3b )(2a3b ) x =2a+3b
  24. m(nx ) (mn ) (m+x ) = n 2 1 n (2m n 2 3 m 2 n ) mnmx( m 2 +mxmnnx ) = n 2 1 n n (2mn3 m 2 ) mn mx m 2 mx+ mn +nx = n 2 2mn+3 m 2 nx2mx = n 2 2mn+3 m 2 + m 2 x(n2m ) = n 2 2mn+4 m 2 x (n2m ) = (n2m ) 2 x =n2m

Ejercicio 143

CAPITULO XVI

Ecuaciones literales de primer grado con una icognita
Ejercicio 143
Resolver las siguientes ecuaciones:
  1. a(x+1 ) =1 x+1 = 1 a x = 1 a 1 x = 1a a
  2. ax4 =bx2 axbx =42 x(ab ) =2 x = 2 ab
  3. ax+ b 2 = a 2 bx axbx = a 2 b 2 x (ab ) =(a+b )(ab ) x =a+b
  4. 3(2ax ) +ax = a 2 +9 6a3x+ax = a 2 +9 x(a3 ) = a 2 6a+9 x = (a3 ) 2 (a3 ) x =a3
  5. a(x+b ) +x(ba ) =2b(2ax ) ax +ab+bx ax =4ab2bx bx+2bx =4abab 3b x = 3 a b x =a
  6. (xa ) 2 (x+a ) 2 =a(a7x ) [(xa ) +(x+a ) ] [(xa ) (x+a ) ] = a 2 7ax (x a +x+ a ) ( x a x a ) = a 2 7ax 2x(2a ) +7ax = a 2 4ax+7ax = a 2 3ax = a 2 x = a 2 3 a x = a 3
  7. axa(a+b ) =x(1+ab ) ax+x =a(a+b ) 1ab x(a+1 ) = a 2 + ab 1 ab x(a+1 ) = a 2 1 x (a+1 ) = (a+1 )(a1 ) x =a1
  8. a 2 (ax ) b 2 (xb ) = b 2 (xb ) a 2 (ax ) b 2 (xb ) b 2 (xb ) =0 a 2 (ax ) 2 b 2 (xb ) =0 a 3 a 2 x2 b 2 x+2 b 3 =0 a 3 +2 b 3 = a 2 x+2 b 2 x a 3 +2 b 3 =x( a 2 +2 b 2 ) x = a 3 +2 b 3 a 2 +2 b 2
  9. (x+a ) (xb ) (x+b ) (x2a ) =b(a2 ) +3a x 2 bx+axab( x 2 2ax+bx2ab ) =ab2b+3a x 2 bx+axab x 2 +2axbx+2ab =ab2b+3a 3ax2bx+ ab = ab 2b+3a x (3a2b ) = 3a2b x =1
  10. x 2 + a 2 = (a+x ) 2 a(a1 ) x 2 + a 2 = a 2 +2ax+ x 2 a 2 +a a 2 a =2ax a(a1 ) =2ax a (a1 ) 2 a =x x = a1 2
  11. m(nx ) m(n1 ) =m(mxa ) m [(nx ) (n1 ) ] = m (mxa ) n x n +1 =mxa a+1 =mx+x a+1 =x(m+1 ) x = a+1 m+1
  12. xa+2 =2ax3(a+x ) 2(a5 ) xa+2 =2ax3a3x2a+10 x2ax+3x =5a+a+102 4x2ax =84a 2 x (2a ) = (2a ) x =2
  13. a(xa ) 2bx =b(b2ax ) ax a 2 2bx = b 2 2abbx ax2bx+bx = a 2 2ab+ b 2 axbx = (ab ) 2 x(ab ) = (ab ) 2 x = (ab ) 2 ab x =ab
  14. ax+bx = (x+ab ) 2 (x2b ) (x+2a ) ax+bx = x 2 +2ax+ a 2 2ab2bx+ b 2 ( x 2 +2ax2bx4ab ) ax+bx = x 2 + 2ax + a 2 2ab 2bx + b 2 x 2 2ax + 2bx +4ab x(a+b ) = a 2 +2ab+ b 2 x = (a+b ) 2 a+b x =a+b
  15. x(a+b ) 3a(a2 ) =2(x1 ) x(ab ) x(a+b ) 2(x1 ) +x(ab ) =3+a(a2 ) ax+ bx 2x+2+ax bx =3+ a 2 2a 2ax2x = a 2 2a+32 2x(a1 ) = (a1 ) 2 x = (a1 ) 2 2 (a1 ) x = a1 2
  16. (m+4x ) (3m+x ) = (2xm ) 2 +m(15xm ) 3 m 2 +mx+12mx+ 4 x 2 = 4 x 2 4mx+ m 2 +15mx m 2 3 m 2 +13mx =11mx 13mx11mx =3 m 2 2mx =3 m 2 x = 3 m 2 2 m x = 3m 2
  17. a 2 (ax ) a 2 (a+1 ) b 2 (bx ) b(1 b 2 ) +a(1+a ) =0 a 2 (ax ) b 2 (bx ) = a 2 (a+1 ) a(1+a ) +b(1 b 2 ) a 3 a 2 x b 3 + b 2 x =a(a+1 ) [a1 ] +b b 3 a 3 a 2 x+ b 2 x =a( a 2 1 ) +b a 3 +( b 2 a 2 ) x = a 3 a+b x = ba b 2 a 2 x = ba (b+a )(ba ) x = 1 a+b
  18. (axb ) 2 =(bxa ) (a+x ) x 2 (b a 2 ) + a 2 +b(12b ) (axb ) 2 (bxa ) (a+x ) + x 2 (b a 2 ) = a 2 +b(12b ) a 2 x 2 2abx+ b 2 abx b x 2 + a 2 +ax+ b x 2 a 2 x 2 = a 2 +b2 b 2 3abx+ax =b2 b 2 b 2 ax (13b ) =b (13b ) x = b a
  19. (x+b ) 2 (xa ) 2 (a+b ) 2 =0 (x+b ) 2 (xa ) 2 = (a+b ) 2 [(x+b ) +(xa ) ] [(x+b ) (xa ) ] = (a+b ) 2 (x+b+xa ) ( x +b x +a ) = (a+b ) 2 (2x+ba )(a+b ) = (a+b ) 2 2x+ b a =a+ b 2x =a+a 2 x = 2 a x =a
  20. (x+m ) 3 12 m 3 = (xm ) 3 +2 x 3 (x+m ) 3 + (xm ) 3 =2 x 3 +12 m 3 [(x+m ) +(xm ) ] [ (x+m ) 2 (x+m ) (xm ) + (xm ) 2 ] =2( x 3 +6 m 3 ) (x+ m +x m ) [ x 2 + 2mx + m 2 ( x 2 m 2 ) + x 2 2mx + m 2 ] =2( x 3 +6 m 3 ) 2 x[2 x 2 +2 m 2 x 2 + m 2 ] = 2 ( x 3 +6 m 3 ) x( x 2 +3 m 2 ) = x 3 +6 m 3 x 3 +3x m 2 = x 3 +6 m 3 x = m 3 3 m 2 x =2m