Question 202701: Solve the following problems for X,Y and Z
X+Y+2Z=5
X+Y=5
X+3Y+Z=11
Answer by PRMath(133)   (Show Source):
You can put this solution on YOUR website! Solve the following problems for X,Y and Z
X+Y+2Z=5
X+Y=5
X+3Y+Z=11
This is solving a system inthree variables. So, you should look at your equations above and see that one of the equations has already eliminated a "Z" variable. So, let's choose two equations and eliminate the "z".
x + y + 2z = 5
x + 3y + 1z = 11 (To eliminate the "Z", let's multiply this equation by -2)
NEW equations:
x + y + 2z = 5
-2x -6y -2z = -22
_____________________________ (add these two equations)
-1x -5y = -17 Let's use this equation with the equation above that didn't have a "z" variable.
-1x -5y = -17
x + y = 5
______________(let's add these two equations because I can already see that the
-4y = -12 "X" variables are going to be eliminated when I add both equations
Divide both sides by -4
y = 3
Now we know that y = 3, so let's "plug" that into one of our original equations:
If y = 3, then we plug in like this:
x + y = 5
x + 3 = 5
x = 5 - 3
x = 2
Now we know that y = 3 and x = 2. Let's plug that info into another equation
x + y + 2z = 5
2 + 3 + 2z = 5
5 + 2z = 5
2z = 5 - 5
2z = 0
z = 0
Now we have: x = 2, y = 3 and z = 0. Does this check out? Let's use all our original equations and plug in the numbers.
X+Y+2Z=5
2+3+2(0) = 5
5 = 5
YES. The numbers work for the first equation. Let's check another.
X+Y=5
2 + 3 = 5
5 = 5
YES. The numbers work for the 2nd equation. One more check....
X+3Y+Z=11
2 + 3(3) + 0 = 11
2 + 9 = 11
11 = 11
YES. The numbers work for the 3rd equation. We have successfully solved a system in 3 variables.
For extra info:
To solve a system in 3 variables:
Select ANY two of the 3 equations and work to get one equation in two variables.
THEN... choose a different pair of equations and eliminate the SAME variable.
You will then have two equations and can solve for one of the variables. Use that variable and plug it into the other "two variable" equation.
Now you'll have the answer for two of the variables and.....can easily use an original equation to plug in the two answers to get your third.
I hope this helps. :-)
|
|
|
