Right now we're doing systems of Linear Equations in Three Variables. It says to solve the following system:
x - y + 3z = 4
x + 2y - z = -3
4x + 3y + 2z = -5
Whatever I try with this problem (and the others like it) it comes out wrong. I've had about one success.
And Is there a certain order you're supposed to solve them in? Or does that not make a difference?
I will assume you are using elimination, although there are several other methods.
In elimination, you first have to decide which variable you want to eliminate. Usually that is determined by choosing the simplest coefficients, in this case I would go for z
1st: x-y + 3z = 4
2nd times 3: 3x + 6y - 3z = -9
add them 4x + 5y = -5 (#4)
2nd times 2: 2x + 4y - 2z = -6
3rd: 4x + 3y + 2z = -5
add them 6x + 7y = -11 (#5)
#4 times 3 : 12x + 15y = -15
#5 times 2: 12x + 14y = -22
subtract them
y = 7
back into #4 : 4x + 35 = -5
4x = -40
x = -10
back into #1 : -10 - 7 + 3z = 4
3z = 21
z = 7
x = -10 , y = 7, z = 7
When solving a system of linear equations in three variables, there are a few different approaches you can take. One of the most common methods is called the "elimination method" or "substitution method". Let's go through the steps to solve the given system of equations:
Step 1: Choose two equations and eliminate one variable.
We can start by eliminating the variable "x". Looking at the first two equations, we see that if we add them together, the "x" terms will cancel out. So, let's add equation 1 and equation 2:
(x - y + 3z) + (x + 2y - z) = 4 + (-3)
This simplifies to:
2x + y + 2z = 1
Step 2: Now, choose another pair of equations and eliminate the same variable, or a different variable.
Let's eliminate the variable "x" again. This time, we will choose equations 2 and 3. We can multiply equation 2 by 4, and equation 3 by -1, so that the "x" terms will cancel out:
4(x + 2y - z) + (-1)(4x + 3y + 2z) = (-3)(4) + (-5)
Simplifying this equation gives us:
5y + 6z = -7
Step 3: We now have two new equations:
2x + y + 2z = 1 (from Step 1)
5y + 6z = -7 (from Step 2)
We can now solve this new system of equations using either the elimination method or the substitution method. For simplicity, let's use the substitution method.
From the second equation, we can solve for "y" in terms of "z":
5y = -7 - 6z
y = (-7 - 6z)/5
Now, substitute this expression for "y" into the first equation:
2x + (-7 - 6z)/5 + 2z = 1
Simplify and rearrange this equation to solve for "x" in terms of "z":
10x + (-7 - 6z) + 10z = 5
10x + 3z = 12
10x = 12 - 3z
x = (12 - 3z)/10
So now we have expressions for "x" and "y" in terms of "z".
Step 4: Substitute these expressions for "x" and "y" into any of the original equations to find the value of "z". Let's substitute into the first equation:
(12 - 3z)/10 - (-7 - 6z)/5 + 3z = 4
Simplify and solve for "z":
(2(12 - 3z) - (-7 - 6z) + 6z)/10 = 4
(24 - 6z + 7 + 6z + 6z)/10 = 4
(31 + 6z - 6z)/10 = 4
31/10 = 4
This equation is not true for any value of "z". Therefore, the system of equations is inconsistent and does not have a solution.
To answer your second question, there isn't a specific order you must solve the equations in, but choosing the equations strategically can make the process easier or faster. In this case, we eliminated the variable "x" in Steps 1 and 2. However, if you had chosen a different approach, such as eliminating the variable "y" or "z" first, the process would have been different, but the end result (inconsistent system) would have been the same.
Keep in mind that it's common to make calculation errors when solving systems of equations by hand, so double-check your work and consider using a calculator to reduce the chances of mistakes.