1. MasMates.com Álgebra
Colecciones de ejercicios Sistemas de ecuaciones
1. Resuelve el sistema:
2x + y = 4 2x + y = 1 4x + 2y = 1 2x + 5y = 1
1. 2. 3. 4.
x+y=3 x+y=0 x- y=1 x+ y=0
2x + y = 3 2x - y = 2 2x + y = 1 3x + 5y = 2
5. 6. 7. 8.
3x - 2y = 1 3x - 2y = 2 3x + 3y = 1 5x + y = 7
2x + 3y = 7 2x + 3y = 1 3x + 3y = 1 2x + 4y = 5
9. 10. 11. 12.
3x - 2y = 4 3x + 2y = 4 2x - 5y = 3 3x - 5y = 2
3(x+y) = 2y+1 3(x+y) = 1+y 2(x-y) = y+1 2(x+2) = y+3
13. 14. 15. 16.
2(x-y)-1 = 5 5(x-y) = 9-y 3x = 2(y+2) 3(3x+y) = 1-3y
2(x+y) = 2-y 2(x+y) = 2+y 2(x+y) = 3-2x 2(x+y) = 1-y
17. 18. 19. 20.
3(x+2y) = 2 4(x+y) = 7-2x 3(x+1) = 5-y 9(2x+3y) = 7-3y
3x+2(y-2)-1 = 2x-4 x+y-2(x-1) = 1+2y 1-y-2(x+1) = 3y-2 1-2y-2(2x-1) = x+2
21. 22. 23. 24.
3(x-1)-y+3 = 2(x-y) x+2(y+1) = 3-2x 1-x-2(y+1) = 2x-y x-2-2(y-1) = 2x-y
2(x-1)-2(y-1) = 1-y 3(x-1)-2(y-1) = 3-y 3(x+1)-2(y-1) = 4-5y 3(x-1)+2(y+1) = y-5
25. 26. 27. 28.
2(x+1)-2(y+1) = 1-x 2(x+1)-3(y+1) = 1-2y 2(x-1)-2(y+1) = y-1 2(x+1)-2(y-1) = 3-6y
2(x-y+1)+y = 2(y+1)-1 2(x-y)-2(y-5) = 3-y 2(x-y-1)-2(y-1) = 1-(x+y) 2(x+y)+3(y+1) = y+4
29. 30. 31. 32.
3(x-y+2)-y = 2(3-y)-4 2(x+y)-3(x-2) = 2(1-2x) 2(x-2y)-2(y-1) = y+1 5(x+y)-3(x-1) = 5-x
2. Resuelve el sistema:
x 1 x y
x + 2y = 1 2x + y = 2 x + 2y = 3 + y= - = 1
2 6 2 4
1. x y 1 2. x y 2 3. x y 5 4. 5.
+ = + = - = 1 x y
3 2 6 2 6 3 2 3 6 2x + 3y = - =0
6 3 2
x y 1 1 1 1 1 1 1 x y-6 x y-4
+ =- x+ y= x- y= - =3 - =2
3 2 6 3 2 3 3 2 6 6 3 2 4
6. 7. 8. 9. 10.
x 1 1 1 3 3 1 y+4 2-y
+ y= 0 x- y= x- y= x- = 1 x- =0
2 3 4 12 4 2 8 2 4
x y+2 3 x y+1 1 x 4x+3y x 3x-2y 3x 2y+3x 1
+ = + = - =1 - =1 - =
2 3 4 2 3 3 2 12 2 12 2 3 6
11. 12. 13. 14. 15.
x-1 y 1 x+1 y 7 2x+y 2x 3x-y x+2y 5y 3
- = + = x - =1 - =1 - =
4 2 8 2 4 12 4 3 9 2 4 8
3. Resuelve el sistema:
2x+y-z = 1 x+y-z = 1 x+y+z = 2 x-y-z = 0 6x-6y+8z = 3
1. x-2y+2z = 3 2. 2x+2x-3z = 1 3. 2x-2y-z = 2 4. x+2y-5z = 2 5. 3x-6y+6z = 1
3x-2y+z = 2 4x-2y-z = 1 3x+5y+2z = 2 3x-2y-4z = 1 12x-12y+12z = 5
2(x+y)+z = x+1 x+z = y x+2y = z+1 3(x+z) = 1-y 2(x+y) = 1-z
6. 2(x-y)-z = y+1 7. 2(x+z) = 3y-1 8. 3x = 2(y+z) 9. 2(y-z) = 3-x 10. 2(x-z) = 2+y
3(x+z)+2y = y-2 2(y+z) = 3(1-x-z) 3(x+z) = 4(y+1) 2(x-y) = z-1 y+z = 1-4x
x 3y+z x y z x+y+7 z y+6 x-z x+3y z+1
+ = -1 - - =1 - =0 - =1 - =y
3 6 4 8 2 3 2 2 3 6 3
x y+2z x y+z 8 x-z y y+6 x-z x-4z
11. + = -1 12. - = 13. + +1 = 0 14. - =1 15. 2+z+ =y
2 4 3 2 3 4 8 3 6 4
x y-z x-y z z 2x+y 5 x-2y-5 z x-5 z-y
- = -1 - =1 - - =0 + =1 - =y
3 6 6 3 2 6 3 3 2 5 2
4. Resuelve el sistema:
5 de diciembre de 2009 Página 1 de 2

2. MasMates.com Álgebra
Colecciones de ejercicios Sistemas de ecuaciones
x-2y = 1 3x+y = 1 2x-y = 1 2x+3y = 1 2x+3y = 1
1. 2
2. 2
3. 2
4. 2
5.
x +4y = 1 x -y = 3 3x-y = 2 x +2y = 2 2y-x2 = 1
x-y = 3 x-y = 1 2x-y = 2 2x-3y = 0 x-2y = 1
6. 7. 2 2
8. 2 2
9. 2 2
10.
2 2
x +y = 5 x -2y = 2 2x -y = 2 x -2y = 1 2x2-y2 = 1
x2-y = 3 3x-y2 = 2 x2-y = 6 2x+y2 = 2 2x2+y = 3
11. 12. 13. 14. 15.
2 2 2 2 2 2 2 2
x +y = 5 3x -2y = 4 2x +y = 27 4x +6y = 7 16 x -1 +y2 = 1
2
xy-x+y = 4 xy-x+2y = 3 xy+2x+y = 0 xy+x-y = 1 xy-x+y = 1
16. 17. 18. 19. 20.
xy-y = 2 xy+x+3 = 0 x-xy = 2 2xy-x-y = 0 2xy+2x-y = 1
5. Resuelve el sistema:
xy-2y = 1 x2+y2-2x = 1 x2+y2-x-y = 2 2 2
x + y + 2x = 1
2 2
x + y - 2xy = 4
1. 2
2. 3. 4. 5.
x2+y2+2y = 3 x2+y2-2x+y = 2
2 2 2 2
xy+x+y = 2 x - y + 2y = 1 x - y - xy = 5
1 1 3 1 4 1 1 2 1 1
+ = + +1=0 - =1 - =1 +y=1
x y 2 x y x y x y x
6. 7. 8. 9. 10.
2 2 1 1 1 3 2 1
- =1 +y+1 = 0 + =2 - =1 3x - = 4
x y x 3x 2y 2x 3y y
2
y- =1 1 1
x - =3 x-y=1 x+y=2 2x + 3y = 1
11. 12. x y 13. 14. 15.
2 x+2 - y = 1 x+ y-1 = 1 x+2 + y = 1
x- = 1 2
x + y - xy = 1
y
x-1 + y = 2
x+y = 2 x-y=1 3x+y-1 = 0 2x-1 + y+2 = 2
16. 17. 18. 19. 20. 1 1
x+3 - y=1 x+1 - y-1 = 1 2x-1- 2-y+1 = 0 2 2x-1 + y = 1 + =2
x-1 y
Soluciones
1 -1 -1 1 2 -1 3 -1 2 -1 3 1
1.1. (1,2) 1.2. (1,-1) 1.3. , 1.4. , 1.5. (1,1) 1.6. (2,2) 1.7. , 1.8. , 1.9. (2,1) 1.10. (2,-1) 1.11. , 1.12. , 1.13.
2 2 3 3 3 3 2 2 3 3 2 2
1 -1 -2 1 1 1 3 -2 -1 1 1 -1
(1,-2) 1.14. (1,-1) 1.15. (2,1) 1.16. , 1.17. 2, 1.18. ,1 1.19. , 1.20. , 1.21. (-1,1) 1.22. (-1,2) 1.23.
, 1.24. ,
3 3 3 2 2 2 2 3 2 2 3 3
-2 1 -3 1 2 1 3 -1 1 -2 1
1.25. (1,1) 1.26. (2,2) 1.27. , 1.28. , 1.29. (-2,-1) 1.30. (-2,1) 1.31. , 1.32. , 2.1. (-1,1) 2.2. (2,-2) 2.3. 2, 2.4. ,
3 3 2 2 3 3 2 2 2 3 2
1 1 3 2 1 -1 2 1
2.5. (3,2) 2.6. (-2,1) 2.7. , 2.8. , 2.9. (2,-2) 2.10. (1,-2) 2.11. , 2.12. ,-1 2.13. (3,-2) 2.14. (2,3) 2.15. 1, 3.1. (1,2,3) 3.2.
2 3 2 3 2 2 3 2
1 1 1 1 1 -1 1 2
(1,1,1) 3.3. (1,-1,2) 3.4. (3,2,1) 3.5. , , 3.6. (1,1,-2) 3.7. (2,1,-1) 3.8. 1, ,1 3.9. ,1, 3.10. , ,-1 3.11. (-4,0,2) 3.12. (2,-4,0) 3.13.
2 3 4 2 3 3 3 3
3 1 -2 7 -1 5
(0,-4,2) 3.14. (2,0,-4) 3.15. (0,2,-4) 4.1. (1,0), (-3,-2) 4.2. (1,-2), (-4,13) 4.3. (1,1), , 4.4. (2,-1), , 4.5. (-1,4), , 4.6. (2,-1), (1,-2) 4.7.
4 2 3 9 3 9
5 -1 1
(2,1) 4.8. (1,0), (3,4) 4.9. (3,2), (-3,-2) 4.10. (-1,-1), , 4.11. (1,-2), (-1,-2), (2,1), (-2,1) 4.12. (2,2); (2,-2) 4.13. (1,-5), (-1,-5), (3,3), (-3,3) 4.14. ,1 ,
7 7 2
1 -1 -2 1 1
,-1 4.15. (1,1), (-1,1) 4.16. (2,2), -3, 4.17. (-1,2), (-3,0) 4.18. (1,-1), ,4 4.19. (1,1), ,-1 4.20. (-1,-1), ,1 5.1. (1,-1) 5.2. (0,1), (2,-1) 5.3.
2 2 3 3 2
-4 -2 -1 1 1 1 1 1 2 -1 1
(2,1), , 5.4. (0,1), (-2,-1) 5.5. (3,1), (-3,-1) 5.6. (1,2) 5.7. (1,-2), ,2 5.8. , 5.9. , 5.10. 2, , , 5.11. (-1,-1), (2,2) 5.12. ,1
5 5 3 3 2 2 3 2 3 2 4
5.13. (2,1) 5.14. (0,2), (1,1) 5.15. (2,-1) 5.16. (1,1) 5.17. (3,2) 5.18. (1,-2), (5,-14) 5.19. (1,-1) 5.20. (2,1)
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