Explain, in complete sentences, which method you would use to solve the following system of equations and why you chose that method. Provide the solution to the system.
x - 3y + 2z = -12
x + 2y + 3z = 6
2x - 3y - z = -2
I'd use elimination, since the coefficients are small
to eliminate x, subtract the 1st two, subtract twice #1 from #3 and we have
-5y - z = -18
-9y + 5z = -26
now eliminate y or z in the same way, and then plug the values back in to obtain the others.
What strategy did you use to solve 5? Explain the step in complete sentences
To solve the given system of equations, I would use the method of elimination. This method involves manipulating the equations by adding or subtracting them in order to eliminate one variable at a time.
First, we need to eliminate the term with "x". To do this, let's start by adding the second equation to the first equation.
(x - 3y + 2z) + (x + 2y + 3z) = -12 + 6
Simplifying the equation, we get:
2x - y + 5z = -6
Next, we need to eliminate the term with "x" again. To do this, let's multiply the third equation by 2 and add it to the equation we obtained in the previous step.
2(2x - 3y - z) + (2x - y + 5z) = -2 + (-6)
Simplifying the equation, we get:
4x - 6y - 2z + 2x - y + 5z = -8
Combining like terms, we have:
6x - 7y + 3z = -8
Now, we have a new equation:
6x - 7y + 3z = -8
We can now solve the system of equations using any method we prefer. For simplicity, I will continue using the method of elimination.
Next, we need to eliminate the "y" term. To do this, let's multiply the first equation by 7 and subtract it from the equation we obtained earlier.
7(2x - y + 5z) - (6x - 7y + 3z) = 7(-6) - (-8)
Simplifying the equation, we get:
14x - 7y + 35z - 6x + 7y - 3z = -42 + 8
Combining like terms, we have:
8x + 32z = -34
Now, we have a new equation:
8x + 32z = -34
Finally, we can solve this equation for "x" in terms of "z". Dividing both sides of the equation by 8, we get:
x + 4z = -4.25
Now, we can substitute this value of "x" into one of the original equations to solve for the remaining variables. Let's substitute this value into the first equation:
(-4.25) - 3y + 2z = -12
Simplifying the equation, we have:
-3y + 2z = -7.75
We now have a system of two equations with two variables:
-3y + 2z = -7.75 (Equation 1)
x + 4z = -4.25 (Equation 2)
To solve this system further, we can choose another method such as substitution or elimination.
To solve the given system of equations:
x - 3y + 2z = -12 --- Equation 1
x + 2y + 3z = 6 --- Equation 2
2x - 3y - z = -2 --- Equation 3
One possible method to solve this system is the Elimination method. The Elimination method involves manipulating the given equations in such a way that when the equations are added or subtracted, one variable gets eliminated, allowing us to solve for the remaining variables.
In this case, I would solve the system using the Elimination method by first eliminating the variable x. To cancel out the x terms in Equation 1 and Equation 2, we can multiply Equation 2 by -1 and add it to Equation 1. This will give us a new equation without the x term:
(1)*[x - 3y + 2z = -12]
(-1)*[x + 2y + 3z = 6]
Resulting in:
-3y - z = -18 --- Equation 4
Now, we have eliminated the variable x. Let's proceed to eliminate another variable, y. To cancel out the y terms in Equation 3 and Equation 4, we can multiply Equation 4 by 3 and add it to Equation 3. This will provide us with another new equation without the y term:
(3)*[-3y - z = -18]
[2x - 3y - z = -2]
Resulting in:
2x - 7z = -20 --- Equation 5
Now, we can solve Equation 5 for x in terms of z by isolating x:
2x = 7z - 20
x = (7z - 20)/2 --- Equation 6
Now, we substitute Equation 6 into Equation 1, 2, or 3 to solve for either y or z. Let's substitute it into Equation 1:
(7z - 20)/2 - 3y + 2z = -12
Now, we can solve this equation to find the value of y in terms of z.
Once we obtain the values for y and z, we can substitute those values into any of the original equations to find the value of x.