Solving systems of linear equations in three variables.
Ok so i don't really totally get these problems. Can you help me solve this one? and explain everything please?
2x+3y=6+z
x-2y=-1-z
3x+y=-1+3z
put them in standard form (x,y,z)
2x+3y-z=6
x-2y+z=-1
3x+y-3z=-1
You can solve these a number of ways, I will show you one.
add 1st and second equations
3x+y=5
now multiply equation 2 by three, then add equations 2 and 3.
6x-5y=-4 check that.
now on these two remaining equations, multiply the first by 2, then subtract.
7y=14
y=2 work back up to find x, then z.
Ok, this helps a lot, but hold on, i don't get how you got 7y=14? because when i multiply 3x+y=5 by (-2) i get -6x-2y=8. And then when combining that with 6x-5y=-4 i get -7y=4? how did you get 7y=14???
3x+y=5 multiply -2, your way
6x-5y=-4 to get
-6x-2y=-10
6x-5y=-4 add to get
-7y=-14
y=2
Oh wait, i forgot to multiply the 5 by -2! (idk how i got 8, but watever its not that important) so right you would get y=2.
ok i think i get it now! THANKS!
Sure, I can help you solve this system of linear equations with three variables. To start, let's assign variables to each equation:
Equation 1: 2x + 3y = 6 + z
Equation 2: x - 2y = -1 - z
Equation 3: 3x + y = -1 + 3z
We'll use a process called elimination to solve this system. The goal is to eliminate one variable at a time by manipulating the equations in a way that they cancel each other out.
Step 1: Choose two equations and eliminate a variable.
Let's eliminate the variable "z" using equations 1 and 2. To do this, we can add equation 1 and equation 2 together:
(2x + 3y) + (x - 2y) = (6 + z) + (-1 - z)
Combining like terms, we get:
3x + y = 5
We have now eliminated the variable "z" from the system.
Step 2: Choose two different equations and eliminate another variable.
Now, let's eliminate the variable "z" using equations 2 and 3. Multiply equation 2 by 3 and equation 3 by -1, then add them together:
3(x - 2y) + (-1)(3x + y) = 3(-1 - z) + (-1)(-1 + 3z)
Simplifying, we have:
3x - 6y - 3x - y = -3 - 3z + 1 - 3z
Simplifying further:
-7y = -6z - 2
Step 3: Solve for one variable.
We now have two equations:
3x + y = 5
-7y = -6z - 2
Let's solve equation 2 for "y":
-7y = -6z - 2
y = (6z + 2) / -7
Step 4: Substitute the value of the eliminated variable into one of the original equations.
Substitute the value of "y" into equation 1:
3x + (6z + 2) / -7 = 5
Step 5: Solve for the remaining variables.
Now we can solve the equation above for "x" in terms of "z." Multiply everything by -7 to get rid of the fraction:
-21x - 6z - 2 = -35
-21x = -6z + 33
x = (-6z + 33) / -21
Finally, let's substitute the values of "x" and "y" back into equation 3:
3((-6z + 33) / -21) + (6z + 2) / -7 = 5
Simplify the equation above to solve for "z." After finding the value of "z," substitute it back into the expressions we found for "x" and "y" to get their respective values.
Note: The steps provided above can be time-consuming when solving systems of linear equations with three variables. You can also use matrix methods or an online solver to get the solution more quickly.