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.