Final answer:
To solve each system of equations by elimination, we add or subtract the equations to eliminate one variable, and then solve for the remaining variable. The solution varies depending on the system.
Step-by-step explanation:
To solve each system of equations by elimination, we will add or subtract the equations in such a way that one variable cancels out. We will then solve for the remaining variable and substitute it back into one of the original equations to find the value of the other variable.
a. Add the two equations together:
(2x + 3y) + (-2x - 3y) = 16 + (-16)
0 = 0
Since 0 = 0 is always true, this means that the system of equations has infinitely many solutions and the two equations are dependent.
b. Add the two equations together:
(-2x - 3y) + (2x + 3y) = (-16) + 16
0 = 0
Similarly, this system of equations also has infinitely many solutions and the two equations are dependent.
c. Multiply the first equation by 2 and add it to the second equation:
(-2x + 5y) + (2x + 3y) = 5 + 16
8y = 21
y = 2.625
Substitute the value of y back into the first equation to find x:
-2x + 5(2.625) = 5
-2x = 5 - 13.125
-2x = -8.125
x = 4.0625
So the solution to this system of equations is x = 4.0625 and y = 2.625.
d. Multiply the first equation by -5 and add it to the second equation:
(5x + 3y) + (-5x + 3y) = 13 + 0
6y = 13
y = 2.1667
Substitute the value of y back into the first equation to find x:
5x + 3(2.1667) = 13
5x = 13 - 6.5
5x = 6.5
x = 1.3
Therefore, the solution to this system of equations is x = 1.3 and y = 2.1667.
e. Multiply the first equation by -5 and add it to the second equation:
(-10x + 6y) + (-(-10x + 6y)) = 6 + 0
0 = 6
Since 0 = 6 is never true, this system of equations has no solution and the two equations are inconsistent.
f. Multiply the first equation by -2 and add it to the second equation:
(-5x + 2y) + (2x + 2y) = 2 + 2
-3x + 4y = 4
This system of equations has infinitely many solutions since we only obtained one equation after elimination, so the two equations are dependent.