Find the solution of the following system of equation by substitution method, if it has a unique solution. Also, state whether the system inconsistent or not?
1) 4x + 9y = 5 ; 3x – 5y = 39
2) 2x + 5y = 3 ; 6x + 15y = 11
3) 5x – 7y – 21 = 0 ; 25x – 35y = 63
4) 2x + 3y = 17 ; 3x – 2y = 6
Answers
Answer:
1) x =8 y = -3
2) Has no solution, system inconsistent
3) Has no solution, system inconsistent
4) x = 4 y = 3
Step-by-step explanation:
1) 4x + 9y = 5 ; 3x – 5y = 39
L1 4x + 9y = 5
L2 3x – 5y = 39
4x = 5 - 9y
x = (5 - 9y) / 4
Reemplace
3x – 5y = 39
3 (5 - 9y) / 4 - 5y = 39
15/4 - 27y/4 - 5y = 39
- 27y/4 - 5y = 39 - 15/4
(-27y - 20y) / 4 = 141 / 4
-47y / 4 = 141/4
-47y = 141
y = 141/-47
y = -3
x = (5 - 9y) / 4
x = (5 - 9(-3)) / 4
x = (5+27) / 4
x = 32/4
x = 8
2) 2x + 5y = 3 ; 6x + 15y = 11
L1 2x + 5y = 3
L2 6x + 15y = 11
No solution are paralell lines
2x + 5y = 3 multiply by 3
6x + 15y = 9 is similar to 6x + 15y = 11
System inconsistent
9 = 11
3) 5x – 7y – 21 = 0 ; 25x – 35y = 63
L1 5x – 7y = 21
L2 25x – 35y = 63
No solution are paralell lines
5x -7y = 21 multiply by 5
25x - 35y = 105 is similar to 25x – 35y = 63
System inconsistent
105 = 63
4) 2x + 3y = 17 ; 3x – 2y = 6
L1 2x + 3y = 17
L2 3x – 2y = 6
2x + 3y = 17
2x = 17 - 3y
x = (17 - 3y)/2
Reemplace
3x – 2y = 6
3(17 - 3y)/2 - 2y = 6
51/2 - 9y/2 - 2y = 6
- 9y/2 - 2y = 6 - 51/2
(-9y - (2)2y) / 2 = (12-51)/2
(-9y - 4y) / 2 = -39/2
-13y/2 = -39/2
13y = 39
y = 39/13
y = 3
x = (17 - 3y)/2
x = (17 - 3(3))/2
x = (17 - 9)/2
x = 8/2
x = 4