7x-3y+4z=18
13x+6y+8z=30
11x-9y-12z=16
Find x,y,z please show the solution pleaseee
double the 1st: 14x - 6y + 8z = 36
using the 2nd: 13x + 6y + 8z = 30
add them
27x + 16y = 66 , #4
triple the 1st: 21x - 9y + 12z = 54
using the 3rd: 11x - 9y - 12z = 16
subtract them: 10x + 24z = 38
5x + 12z = 19 , #5
#4 times 3 ---> 81x + 48y = 198
#5 times 4 ---> 20x + 48y = 76
subtract:
61x = 122
x = 2
back into #5:
10+12z = 19
z = 9/12 = 3/4
back into #1
14 - 3y + 3 = 18
y = -1/3
notice, my method is not unique. There are several ways this could have been done using elimination.
I usually look over my numbers and decide which coefficients are the simplest to eliminate
I chose the y's because of the simple 3, 6 , 9 combination.
My second choice would have been the z's, since 4,8,12 are also nice multiples
My last choice would have been the x's.
7, 13, 11 are all prime, which usually is not so good in these types of problems
To find the values of x, y, and z, we can use the method of elimination or substitution. Let's use the method of elimination.
Step 1: Multiply the first equation by 2 and the second equation by 3 to make the coefficients of x in both equations the same:
2(7x - 3y + 4z) = 2(18)
3(13x + 6y + 8z) = 3(30)
This simplifies to:
14x - 6y + 8z = 36
39x + 18y + 24z = 90
Step 2: Multiply the first equation by 3 and the third equation by 2 to make the coefficients of x in both equations the same:
3(7x - 3y + 4z) = 3(18)
2(11x - 9y - 12z) = 2(16)
This simplifies to:
21x - 9y + 12z = 54
22x - 18y - 24z = 32
Step 3: Rewrite the system of equations:
14x - 6y + 8z = 36 -> Equation A
39x + 18y + 24z = 90 -> Equation B
21x - 9y + 12z = 54 -> Equation C
22x - 18y - 24z = 32 -> Equation D
Step 4: Add Equation A and Equation D:
(A) + (D):
14x - 6y + 8z + 22x - 18y - 24z = 36 + 32
36x - 24y - 16z = 68 -> Equation E
Step 5: Multiply Equation C by 2 and subtract Equation E:
2(C) - (E):
42x - 18y + 24z - 36x + 24y + 16z = 108 - 68
6x + 42z = 40 -> Equation F
Step 6: Divide Equation F by 6:
6x + 42z = 40
x + 7z = 6 -> Equation G
Step 7: Solve Equation G for x:
x = 6 - 7z -> Equation H
Step 8: Substitute Equation H into Equation C:
21(6 - 7z) - 9y + 12z = 54
126 - 147z - 9y + 12z = 54
Simplify to:
-135z - 9y = -72
15z + y = 8 -> Equation I
Step 9: Rewrite Equation I as:
y = 8 - 15z -> Equation J
Step 10: Substitute Equation H and Equation J into Equation B:
39(6 - 7z) + 18(8 - 15z) + 24z = 90
234 - 273z + 144 - 270z + 24z = 90
Simplify to:
-519z = -288
z = 288/519
z ≈ 0.555
Step 11: Substitute the value of z into Equation H:
x = 6 - 7(0.555)
x ≈ 6 - 3.885
x ≈ 2.115
Step 12: Substitute the values of z and x into Equation J:
y = 8 - 15(0.555)
y ≈ 8 - 8.325
y ≈ -0.325
Therefore, the solution to the system of equations is approximately:
x ≈ 2.115, y ≈ -0.325, and z ≈ 0.555.
To find the values of x, y, and z in the given system of equations, we can use the method of elimination or substitution to solve them simultaneously. Let's use the elimination method in this case.
Step 1: Multiply the first equation by 13, the second equation by 7, and the third equation by -11 to make the coefficients of x in all equations the same:
91x - 39y + 52z = 234
91x + 42y + 56z = 210
-121x + 99y + 132z = -176
Step 2: Now, subtract the second equation from the first equation and the third equation:
(91x - 39y + 52z) - (91x + 42y + 56z) = 234 - 210
-121x + 99y + 132z - (91x + 42y + 56z) = -176 - 210
This gives us:
-81y - 4z = 24
-212x + 57y + 76z = -386
Step 3: Multiply the first equation by 57 and the second equation by -81 to make the coefficients of y in both equations the same:
-81y - 4z = 24
-81y + 456z = 31326
Step 4: Subtract the first equation from the second equation:
(-81y + 456z) - (-81y - 4z) = 31326 - 24
Simplifying, we have:
460z = 31350
z = 31350 / 460
z ≈ 68.15
Step 5: Now that we have the value of z, we will substitute it back into one of the previous equations to find the value of y. Let's use the first equation:
7x - 3y + 4z = 18
Replacing z with its value, we get:
7x - 3y + 4(68.15) = 18
7x - 3y + 272.6 = 18
7x - 3y = -254.6
Step 6: Multiply the second equation by 7 and the third equation by -3 to make the coefficients of x in both equations the same:
91x + 42y + 56z = 210
-33x + 27y + 36z = -48
Step 7: Subtract the second equation from the third equation:
(-33x + 27y + 36z) - (91x + 42y + 56z) = -48 - 210
Simplifying, we have:
-124x - 15y = -258
Step 8: Multiply the second equation by 124 and the first equation by -184 to make the coefficients of y in both equations the same:
-12928x - 1508y = -31992
-12928x + 552y = 46899
Step 9: Subtract the first equation from the second equation:
(-12928x + 552y) - (-12928x - 1508y) = 46899 - (-31992)
Simplifying, we have:
2060y = 78891
y = 78891 / 2060
y ≈ 38.28
Step 10: With the values of y and z, we can substitute them back into the equation from Step 6 to find the value of x:
7x - 3(38.28) + 4(68.15) = 18
7x - 114.84 + 272.6 = 18
7x + 157.76 = 18
7x = 18 - 157.76
7x = -139.76
x = -139.76 / 7
x ≈ -19.97
Therefore, the solution to the system of equations is approximately:
x ≈ -19.97
y ≈ 38.28
z ≈ 68.15