Final answer:
To solve the system of equations by elimination, we multiply and add the equations to eliminate one variable, then solve for the remaining variables.
Step-by-step explanation:
To solve the system of linear equations by elimination, we need to eliminate one of the variables by adding or subtracting the equations. Let's use the first and second equations:
Multiply the first equation by 3 and the second equation by -3 to make the coefficients of x the same:
-9x - 3y - 9z = -24x + 6y + 6z = 30
Add the two equations together:
(-9x - 3y - 9z) + (-24x + 6y + 6z) = (-24 + 30)
-33x + 3y - 3z = 6
Now let's use the first and third equations:
Multiply the first equation by 1 and the third equation by 3 to make the coefficients of x the same:
-3x - y - 3z = 3x - 9y + 3z = -36
Add the two equations together:
(-3x - y - 3z) + (3x - 9y + 3z) = (-36 - 12)
-10y = -48
Divide both sides by -10:
y = 4.8
Now substitute the value of y into any of the original equations to solve for x and z. For example, substitute y = 4.8 into the first equation:
-3x - 4.8 - 3z = -8
-3x - 3z = 3.2
This system does not have unique solutions, but the solution can be represented in terms of x and z.