Solve:
2x+y+z=-2
2x-y+3z=6
3x-5y+4z=7
To solve the system of equations, we can use the method of elimination or substitution. I'll explain how to solve these equations using the elimination method.
Step 1: Choose two equations and eliminate one variable.
Let's choose the first and second equations to eliminate the variable 'x'.
Multiply the first equation by 2 and the second equation by -2:
(2) * (2x + y + z) = (2) * (-2) Simplify: 4x + 2y + 2z = -4
(-2) * (2x - y + 3z) = (-2) * (6) Simplify: -4x + 2y - 6z = -12
Now, add these two equations together:
4x + 2y + 2z + (-4x + 2y - 6z) = -4 + (-12)
Simplify: 0x + 4y - 4z = -16
Simplify further: 4y - 4z = -16
Step 2: Choose two other equations and eliminate the same variable or choose a different variable. Let's eliminate 'x' again by choosing the second and third equations.
Multiply the second equation by 3 and the third equation by 2:
(3) * (2x - y + 3z) = (3) * (6) Simplify: 6x - 3y + 9z = 18
(2) * (3x - 5y + 4z) = (2) * (7) Simplify: 6x - 10y + 8z = 14
Now, subtract these two equations:
(6x - 3y + 9z) - (6x - 10y + 8z) = 18 - 14
Simplify: 0x + 7y + z = 4
Simplify further: 7y + z = 4
Step 3: Set up a system of two equations with two variables to solve.
Now, we have the following system of equations:
4y - 4z = -16 ...(Equation 1)
7y + z = 4 ...(Equation 2)
We can now solve this system using the elimination or substitution method. Since the coefficient of 'z' in Equation 1 is -4, and in Equation 2 it is +1, we can eliminate 'z' by multiplying Equation 2 by 4 and adding it to Equation 1.
(4) * (7y + z) = (4) * (4) Simplify: 28y + 4z = 16
Add this equation to Equation 1:
(4y - 4z) + (28y + 4z) = -16 + 16
Simplify: 32y = 0
Divide both sides by 32:
y = 0
Step 4: Solve for the remaining variables.
Now that we have obtained the value of 'y' as 0, we can substitute this value back into one of the original equations. Let's use Equation 2:
7(0) + z = 4
0 + z = 4
z = 4
Finally, substitute the values of 'y' and 'z' back into one of the original equations to solve for 'x'. Let's use the first equation:
2x + (0) + 4 = -2
2x + 4 = -2
2x = -6
x = -3
Therefore, the solution to the system of equations is:
x = -3
y = 0
z = 4