Using complete sentences, explain which method you would use to solve the following system of equations and why. In your answer, include the solution to one of the variables and how you found it using the method you chose.
2x + y + z = –7
x – 3y + 4z = –14
x – 2y – 3z = –11
To solve the given system of equations, I would use the method of elimination. This method involves getting rid of one variable by adding or subtracting the equations in a way that cancels out the corresponding coefficients of that variable.
First, let's eliminate the variable "x" from the second and third equations. To do this, we multiply the second equation by the coefficient of "x" in the third equation, which is 1. This results in the equation:
x - 3y + 4z = -14 (Equation 2)
x - 2y - 3z = -11 (Equation 3)
Next, we subtract Equation 3 from Equation 2, which eliminates the variable "x":
(x - 3y + 4z) - (x - 2y - 3z) = -14 - (-11)
x - 3y + 4z - x + 2y + 3z = -14 + 11
-y + 7z = -3 (Resulting equation after elimination)
Now, we have two equations. The first equation is: 2x + y + z = -7 (Equation 1) and the second equation is: -y + 7z = -3.
To solve these equations simultaneously, we can use the method of substitution or elimination again. For this explanation, let's use the method of substitution.
From the second equation, we can express "-y" in terms of "z":
-y = -3 - 7z
y = 7z - 3
We can substitute this expression for "y" into the first equation:
2x + (7z - 3) + z = -7
2x + 8z - 3 = -7
2x + 8z = -4
Now, we have two equations:
2x + 8z = -4 (Equation 4)
-y + 7z = -3 (Equation 5)
To find the solution for one of the variables, let's find the value of "z" using Equation 5:
-y + 7z = -3
Substituting "y = 7z - 3" into this equation, we get:
-(7z - 3) + 7z = -3
-7z + 3 + 7z = -3
3 = -3
Since this equation is false, there is no solution for the variable "z" in this system of equations. Therefore, the system is inconsistent, and there are no values for "x", "y", and "z" that satisfy all three equations simultaneously.
In summary, I would use the method of elimination to solve the given system of equations. However, after eliminating one variable and simplifying the equations, we find that the system is inconsistent, meaning there is no solution.