Solve the system of linear equations
x − 2y + 3z = 7
4x + 5y + z = −1
−x + y − 2z = −4
What is the value of z, y, and x?
Add:
x-2y+3z = 7
-x+y-2z = -4
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-y+z = 3
Add #2 and 4*#3:
4x+5y+z = -1
-4x+4y-8z = -16
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9y-7z = -17
Now you know that z = y+3,so
9y-7(y+3) = -17
9y-7y-21 = -17
2y = 4
y = 2
so, z = 5, x = -4
To solve this system of linear equations, we can use the method of elimination or substitution. Here, we will use the method of elimination.
Step 1: Multiply the first equation by 4, the second equation by -1, and the third equation by 1. This gives us:
4x - 8y + 12z = 28 (equation 1)
-4x - 5y - z = 1 (equation 2)
-x + y - 2z = -4 (equation 3)
Step 2: Add equations 1 and 2 to eliminate x. Adding the equations results in:
-13y + 11z = 29 (equation 4)
Step 3: Add equations 3 and 4 to eliminate x. Adding the equations results in:
-12y + 9z = 25 (equation 5)
Step 4: Multiply equation 5 by 3 to simplify the coefficients of y and z. This gives us:
-36y + 27z = 75 (equation 6)
Step 5: Add equation 6 with equation 4 to eliminate y. Adding the equations results in:
38z = 104 (equation 7)
Step 6: Divide equation 7 by 38 to solve for z. This gives us:
z = 104/38
z = 2.74 (approximately)
Step 7: Substitute the value of z back into equation 5 to solve for y. Using equation 5:
-12y + 9(2.74) = 25
-12y + 24.66 = 25
-12y = 25 - 24.66
-12y = 0.34
y = 0.34/-12
y = -0.0283 (approximately)
Step 8: Substitute the values of y and z into any of the original equations to solve for x. Using equation 1:
x - 2(-0.0283) + 3(2.74) = 7
x + 0.0566 + 8.22 = 7
x + 8.276 = 7
x = 7 - 8.276
x = -1.276 (approximately)
Therefore, the solution to the system of linear equations is x = -1.276, y = -0.0283, and z = 2.74.
To solve the system of linear equations, we can use the method of elimination or substitution. Let's use the method of elimination.
Step 1: Multiply the first equation by 4, the second equation by -1, and the third equation by -1 to make the coefficients of x in the first and second equations opposite.
4(x - 2y + 3z) = 4(7) -> 4x - 8y + 12z = 28
-1(4x + 5y + z) = -1(-1) -> -4x - 5y - z = 1
-1(-x + y - 2z) = -1(-4) -> x - y + 2z = 4
The system of equations becomes:
4x - 8y + 12z = 28
-4x - 5y - z = 1
x - y + 2z = 4
Step 2: Add the equations to eliminate x.
(4x - 8y + 12z) + (-4x - 5y - z) + (x - y + 2z) = 28 + 1 + 4
Combine like terms:
-2y + 14z = 33
Step 3: Now we have a new equation -2y + 14z = 33. Let's call it Equation 4.
Step 4: Continue the elimination process by combining Equation 4 with the second equation (which will eliminate y).
(-2y + 14z) + (4x + 5y + z) = 33 + (-1)
Combine like terms:
4x + 12z = 32
Step 5: Now we have another new equation 4x + 12z = 32. Let's call it Equation 5.
Step 6: Finally, combine Equation 5 with the third equation to eliminate x.
(4x + 12z) + (-x + y - 2z) = 32 + (-4)
Combine like terms:
11z + y = 28
Step 7: Now we have another new equation 11z + y = 28. Let's call it Equation 6.
Step 8: Now we have a system of two equations with two variables:
Equation 4: -2y + 14z = 33
Equation 6: 11z + y = 28
We can solve this system using any method, such as substitution or elimination, but let's use substitution.
Step 9: Solve Equation 6 for y:
y = 28 - 11z
Step 10: Substitute the value of y in Equation 4:
-2(28 - 11z) + 14z = 33
Simplify:
-56 + 22z + 14z = 33
Combine like terms:
36z = 89
Step 11: Solve for z:
z = 89/36
Step 12: Substitute the value of z back into Equation 6 to find y:
11(89/36) + y = 28
Simplify:
979/36 + y = 28
Subtract 979/36 from both sides:
y = 28 - 979/36
Simplify:
y = (1008 - 979)/36
y = 29/36
Step 13: Substitute the values of y and z back into Equation 4 to find x:
-2(29/36) + 14(89/36) = 33
Simplify:
-58/36 + 1246/36 = 33
Combine like terms:
1188/36 = 33
Divide both sides by 33:
x = 33
Therefore, the solution to the system of linear equations is x = 33, y = 29/36, and z = 89/36.