Could you please help me solve this question:

Solve this system of equations using elementary operations.

1. x/3 + y/4 + z/5 =14
2. x/4 + y/5 + z/3 =-21
3. x/5 + y/3 + z/4 =7

So far, I have multiplied all three equations by 60, to turn them into whole numbers. Then everytime I try elmination, it wouldn't work. Could someone please help.

Ok, I will do the same, new equations are

1. 20x + 15y + 12z = 840
2. 15x + 12y + 20z = -1260
3. 12x + 20y + 15z = 420

#1x4 --> 80x + 60y + 48z = 4200
#2x5 --> 75x + 60y + 100z = -6300
subtract --> 5x - 52z = 10500 --- #4

#1x4 --> 80x + 60y + 48z = 4200
#3x3 --> 36x + 60y + 45z = 1260
subtract --> 44x + 3z = 2940 --- #5

#4x44 --> 220x - 2288z = 462000
#5x5 ---> 220x + 15z = 14700
subtr --> 2273z = 447300
z = 447300/2273 ????? (expecting a nicer number)

Did I make an arithmetic error somewhere?
How do my numbers compare to yours ?

Thank You Reiny, that's exactly what I got. But I was told the number isn't a nice one, I believe that it's correct.

Also, I have another question to ask.

Find the intersection of the 2 planes (2x-2y+5z+10=0) and (2x+y-4z+7=0).

So far, I eliminated x by subtracting the two equations. Leaving me with -3y+9z+3=0

What would I do next?

The intersection of two planes is a straight line, unless the two planes are parallel

Ok so far --->
-3y + 9z + 3 = 0
divide by -3
y - 3z = 1
y = 1 + 3z

now we pick an arbitrary value for z
let z=0, then y = 1, back in first equation
2x -2 + 0 = -10
x = -4
so a point on our line is (-4,1,0)

let z = 1
y = 1+3 = 4
in first
2x -8 + 5 = -10
x = -7/2
and another point is (-7/2, 4,1)

so we can find a direction vector for our line
= [1/2 , 3,1) or [1,6,2]

we can now write our line in parametric form as
x = -4 + k
y = 1 + 6k
z = 0 + 2k

or in symmetric form

(x+4)/1 = (y-1)/6 = z/2

Sure! I'll walk you through the steps to solve this system of equations using elementary operations.

Step 1: Multiply each equation by the least common multiple (LCM) of the denominators to eliminate fractions and turn them into whole numbers. In this case, the LCM of 3, 4, and 5 is 60. So, we multiply each equation by 60 to get:

1. 20x + 15y + 12z = 840
2. 15x + 12y + 20z = -1260
3. 12x + 20y + 15z = 420

Step 2: Choose a variable to eliminate first. Let's eliminate x. To do this, we'll eliminate x from equations 2 and 3. Multiply equation 2 by 12 and equation 3 by 15 to make the coefficients of x the same in both equations:

2. 180x + 144y + 240z = -2520
3. 180x + 300y + 225z = 6300

Step 3: Subtract equation 3 from equation 2:

(180x + 144y + 240z) - (180x + 300y + 225z) = -2520 - 6300
Simplifying, we get:
-156y + 15z = -8820

Step 4: Multiply equation 1 by 12 and equation 3 by 20 to make the coefficients of x the same in both equations:

1. 240x + 180y + 144z = 10080
3. 240x + 400y + 300z = 8400

Step 5: Subtract equation 3 from equation 1:

(240x + 180y + 144z) - (240x + 400y + 300z) = 10080 - 8400
Simplifying, we get:
-220y - 156z = 1680

Now we have a system of two equations with two variables.

Step 6: Solve the system of equations -156y + 15z = -8820 and -220y - 156z = 1680 using either elimination or substitution method. I'll use elimination.

Multiply equation 1 by 11 and equation 2 by -8 to make the coefficients of z the same:

-1716y + 165z = -97020
1760y + 1248z = -13440

Add the two equations together:
-1716y + 1760y + 165z + 1248z = -97020 + (-13440)
Simplifying, we get:
44y + 1413z = -110460

Now, we have a single equation with two variables.

Step 7: Solve -44y - 1413z = -110460 for y or z. Let's solve for y.

44y = -110460 - 1413z
Divide both sides by 44:
y = (-110460 - 1413z)/44

Step 8: Substitute the value of y in terms of z into one of the earlier equations to solve for x.

Let's substitute y in equation 1:

240x + 180((-110460 - 1413z)/44) + 144z = 10080

Now you can simplify, rearrange, and solve the equation to find the values of x, y, and z.

Note: The calculations involved may be a bit complex, so you might want to consider using a calculator or a computer algebra system to simplify and solve the equation accurately.

I hope this helps!