x+2y+5z=-1
2x-y+z=z
3x+4y-4z=14
solve the system using linear combination method.
Are you sure Eq2 = Z?
To solve the system of linear equations using the linear combination method, we need to eliminate one variable at a time from the equations. Here's how you can do it step by step:
Step 1: Rearrange the equations to align the variables and constants.
Equation 1: x + 2y + 5z = -1
Equation 2: 2x - y + z = z
Equation 3: 3x + 4y - 4z = 14
Step 2: Simplify equation 2 by canceling out z on both sides.
2x - y + z = z
2x - y = 0
Now we have:
Equation 1: x + 2y + 5z = -1
Equation 2: 2x - y = 0
Equation 3: 3x + 4y - 4z = 14
Step 3: Multiply equation 2 by 2 and add it to equation 1 to eliminate the x variable.
2(2x - y) = 0
4x - 2y = 0
Now we have:
Equation 1: x + 2y + 5z = -1
Equation 2: 4x - 2y = 0
Equation 3: 3x + 4y - 4z = 14
x + 2y + 5z = -1
4x - 2y = 0
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5x + 5z = -1 (equation 4)
Step 4: Multiply equation 2 by 3 and add it to equation 3 to eliminate the x variable.
3(4x - 2y) = 0
12x - 6y = 0
Now we have:
Equation 1: x + 2y + 5z = -1
Equation 2: 12x - 6y = 0
Equation 3: 3x + 4y - 4z = 14
12x - 6y = 0
3x + 4y - 4z = 14
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15x - 4z = 14 (equation 5)
Step 5: Now we have a system of two equations with two variables (equations 4 and 5). Solve this system using any method you prefer, such as substitution, elimination or matrices.
Equation 4: 5x + 5z = -1
Equation 5: 15x - 4z = 14
Let's solve the system using elimination method:
Multiply equation 4 by 3 and equation 5 by -1:
15x + 15z = -3
-15x + 4z = -14
----------------
19z = -17
Divide both sides of equation 19z = -17 by 19:
z = -17/19
Substitute z = -17/19 back into equation 4:
5x + 5(-17/19) = -1
5x - 85/19 = -1
Now, solve for x:
5x = -1 + 85/19
5x = -19/19 + 85/19
5x = 66/19
x = 66/19 * 1/5
x = 66/95
Substitute the values of x and z into equation 5:
15x - 4z = 14
15(66/95) - 4(-17/19) = 14
990/95 + 68/19 = 14
990/95 + 340/95 = 14
1330/95 = 14
Simplify further:
14 = 14
Both sides are equal, which means the solution is valid.
Therefore, the solution to the given system of equations is:
x = 66/95, y = unknown, z = -17/19.