(3; - 1) x + z = 4+y x + (y - 3) - 5 = 0
To solve this system of equations, let's first rewrite the second equation in terms of x, y, and z.
x + (y - 3) - 5 = 0 x + y - 3 - 5 = 0 x + y - 8 = 0
Now we have two equations:
x + z = 4 + y x + y - 8 = 0
To solve these equations, we can use substitution or elimination method.
Method 1: Substitution method
From the first equation, we can express x in terms of y and z:
x = 4 + y - z
Substitute this expression for x in the second equation:
(4 + y - z) + y - 8 = 0 5 + 2y - z = 0
Now we have a new equation:
5 + 2y - z = 0 --> z = 5 + 2y
Substitute this expression for z in the first equation:
x + (5 + 2y) = 4 + y
Simplify:
x + 5 + 2y = 4 + y
Combine like terms:
x + y + 5 = 4 + y
The y term cancels out:
x + 5 = 4
Solve for x:
x = 4 - 5 x = -1
So, we have found that x = -1.
To find the values of y and z, substitute the value of x = -1 in one of the original equations:
x + z = 4 + y
-1 + z = 4 + y
Rearrange the equation to solve for z:
z = 4 + y + 1 z = 5 + y
Now, we have the relationship between z and y. We can choose an arbitrary value for y, and find the corresponding value of z.
Let's choose y = 0:
z = 5 + 0 z = 5
So, if y = 0, then z = 5.
Therefore, the solution to the system of equations is x = -1, y = 0, and z = 5.
Method 2: Elimination method
To solve this system using the elimination method, we can eliminate one variable by adding or subtracting the equations.
Let's eliminate the x variable.
x + z = 4 + y --> x = 4 + y - z
Substitute this expression for x in the second equation:
(4 + y - z) + y - 8 = 0 5 + 2y - z = 0
Now we have a new equation:
5 + 2y - z = 0
Rearrange the equation to get z in terms of y:
z = 5 + 2y
Substitute this expression for z in the first equation:
x + (5 + 2y) = 4 + y
Simplify:
x + 5 + 2y = 4 + y
Combine like terms:
x + y + 5 = 4 + y
The y term cancels out:
x + 5 = 4
Solve for x:
x = 4 - 5 x = -1
So, we have found that x = -1.
To find the values of y and z, substitute the value of x = -1 in one of the original equations:
x + z = 4 + y
-1 + z = 4 + y
Rearrange the equation to solve for z:
z = 4 + y + 1 z = 5 + y
Now, we have the relationship between z and y. We can choose an arbitrary value for y, and find the corresponding value of z.
Let's choose y = 0:
z = 5 + 0 z = 5
So, if y = 0, then z = 5.
Therefore, the solution to the system of equations is x = -1, y = 0, and z = 5.
