SOLUTION: Solve the system of equations. X-2y+z=-3 -x+4y=10 2x-y+6z=7 The correct answer is one of the following. Which one? A) no solution B) x=2, y=3, z=1 C) x=-1, y=4,z=2 D)

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Question 1198509: Solve the system of equations.
X-2y+z=-3
-x+4y=10
2x-y+6z=7
The correct answer is one of the following. Which one?
A) no solution
B) x=2, y=3, z=1
C) x=-1, y=4,z=2
D) x=3, y=2, z=-1
E) x=-2, y=3, z=-4
Found 3 solutions by ikleyn, math_tutor2020, MathTherapy:
Answer by ikleyn(52787)   (Show Source): You can put this solution on YOUR website!
.

The most straightforward way is to check given answers by substituting values into equations.

The good strategy is to start from second equation,  since it is shortest.

So,  substitute given values into second equation.

Quickly you will find that values  (B)  fit this equation.

After that,  check values  (B)  for the rest of equations.

Doing this way, you will get right answer quickly and without having any difficulties  (without any headache).


///////////////


After seeing the solutions of other tutors,  it is a mystery for me,
why and for what reason do they agitate for  1000  tons of unnecessary calculations,
when easy and a straightforward way lies on the surface and is just pointed.



Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Let's solve the 2nd equation for x.

-x+4y = 10
-x = 10-4y
x = -10+4y
x = 4y-10

Then plug it into the 1st equation
x-2y+z = -3
4y-10-2y+z = -3
2y+z = -3+10
2y+z = 7

And do the same for the 3rd equation
2x-y+6z = 7
2(4y-10)-y+6z = 7
8y-20-y+6z = 7
7y+6z = 7+20
7y+6z = 27

---------------------------------------------------

We now have this system of equations
2y+z = 7
7y+6z = 27

I'll now isolate z in the 1st equation of this new system.
I'm picking on z because the coefficient here is 1.

2y+z = 7
z = 7-2y

Then substitute it into the other equation and solve for y.
7y+6z = 27
7y+6(7-2y) = 27
7y+42-12y = 27
-5y+42 = 27
-5y = 27-42
-5y = -15
y = -15/(-5)
y = 3
This narrows our choices between choice (B) or choice (E).

Use that value of y to find z
z = 7-2y
z = 7-2*3
z = 7-6
z = 1
We can immediately conclude the answer is choice (B)

If you wanted, let's keep going to find x.
x = 4y-10
x = 4*3-10
x = 12-10
x = 2
This offers further confirmation that (B) is the answer.

If you are in a time crunch, then you can use the multiple choice answers to your advantage.
Simply plug each numeric value in for the corresponding variable. Then simplify each side.

Let's say we wanted to check if (E) was the answer.
We'd plug in x = -2, y = 3, z = -4 into the 1st equation to get...
x-2y+z = -3
-2-2(3)+(-4) = -3
-2-6-4 = -3
-12 = -3
We get a false statement at the end. We need to get the same thing on both sides for it to be a true equation.
Since the last equation is false, it causes the first equation to be false for those x,y,z values.
Therefore, we can rule out choice (E) fairly quickly.
Choices (C) and (D) are eliminated for similar reasoning.

To confirm choice (B) is the solution, plug (x,y,z) = (2,3,1) into each of the original three equations.
You should get true results after simplification.
I'll let you do these confirmations.

---------------------------------------------------
Answer: Choice (B) x=2, y=3, z=1
Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

Solve the system of equations.
X-2y+z=-3
-x+4y=10
2x-y+6z=7
The correct answer is one of the following. Which one?
A) no solution
B) x=2, y=3, z=1
C) x=-1, y=4,z=2
D) x=3, y=2, z=-1
E) x=-2, y=3, z=-4
   x - 2y + z = - 3 ---- eq (i)
 - x + 4y     =  10 ---- eq (ii)
  2x - y + 6z =  7 ----- eq (iii)
       2y + z = 7 ------ Adding eqs (i) & (ii) ----- eq (iv)

 2x - 4y + 2z = - 6 ----- Multiplying eq (i) by 2 ---- eq (v)
      3y + 4z = 13 ------ Subtracting eq (v) from eq (iii) ---- eq (vi)
      8y + 4z = 28 ------ Multiplying eq (iv) by 4 ---- eq (vii)
           5y = 15 ------ Subtracting eq (vi) from eq (vii)
           
At this point, it's quite clear that it's either CHOICE B) or E).
       2(3) + z = 7 ------ Substituting 3 for y in eq (iv)
              z = 7 - 6 = 1 
Now that we've found that y = 3, and z = 1, there's NO DOUBT that CHOICE B) is correct.

If you wish, you can still continue on to determine the value of x - although it's not necessary - by substituting
3 for y in eq (ii), and solving for x. This way however, you will be 100% certain that CHOICE B) is indeed correct.

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