Example 2: Solve the following system of equations:
| 4y - 5 | = | 20 - 3x |
|
| 4x - 7y + 16 | = | 0 |
|
- Rearrange each equation:
3x + 4y = 25
4x - 7y = - 16
- Multiply the first equation by 4 and the second equation by -3:
12x + 16y = 100
-12x + 21y = 48
- Add the equations:
37y = 148
- Solve for the variable:
y = 4
- Plug y = 4 into one of the equations and solve for x:
3x + 4(4) = 25
3x + 16 = 25
3x = 9
x = 3
Thus, the solution to the system of equations is (3, 4).
- Check:
4(4) - 5 = 20 - 3(3) ? Yes.
4(3) - 7(4) = - 16 ? Yes.
Example 3: Solve the following system of equations:
- Rearrange each equation:
2x - 5y = 15
-4x + 10y = 20
- Multiply the first equation by 2:
4x - 10y = 30
-4x + 10y = 20
- Add the equations:
0 = 50
Since
0≠50, this system of equations has no solutions. It is inconsistent (and independent). The equations describe two parallel lines.
Example 4: Solve the following system of equations:
- Rearrange each equation:
6x + 14y = 16
-9x - 21y = - 24
- Multiply the first equation by 3 and the second equation by 2:
18x + 42y = 48
-18x - 42y = - 48
- Add the equations:
0 = 0
Since
0 = 0 for any value of
x, the system of equations has infinitely many solutions. Every ordered pair
(x, y) which satisfies
6x + 14y = 16 (or
-9x - 21y = - 24) is a solution to the system. The system is dependent (and consistent). The two equations describe the same line.
