Solve the system of linear equations and check any solutions algebraically. (If there is no solution, enter NO SOLUTION. If the system is dependent, express x, y, and z in terms of the parameter a.)
3x-3y+6z=6
x+2y-z=8
5x-8y+13z=4
I don't know what parameter A is...
Do I solve for x, y, or z? What is parameter a?
I got the answer (6-a,a,a-2)! Thanks for those who clicked on my questioning with helpful intentions.
In the given system of equations, the parameter "a" is not mentioned. So, we will solve for x, y, and z without considering any parameter.
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 second equation by 3 and the third equation by -1 to make the coefficients of x in all three equations equal. This will help us eliminate the variable x.
The equations become:
1) 3x - 3y + 6z = 6
2) 3x + 6y - 3z = 24
3) -5x + 8y - 13z = -4
Step 2: Subtract equation 1 from equation 2 and equation 1 from equation 3 to eliminate x.
Equations become:
4) 0x + 9y - 9z = 18
5) -5x + 8y - 13z = -10
Step 3: Multiply equation 4 by 5 and equation 5 by 9 to make the coefficients of y in both equations equal.
Equations become:
6) 0x + 45y - 45z = 90
7) -45x + 72y - 117z = -90
Step 4: Add equation 6 and equation 7 to eliminate y.
Equation becomes:
8) -45x + 117z = 0
Step 5: Solve equation 8 for z.
Divide both sides of equation 8 by -9:
-45x + 117z = 0
5x - 13z = 0
We can rewrite equation 8 as:
9) 5x = 13z
Now, we can express x in terms of z using equation 9:
10) x = (13/5)z
Step 6: Substitute the value of x from equation 10 into equation 1 or 2 to express y in terms of z.
Let's use equation 1:
3x - 3y + 6z = 6
Substitute x with (13/5)z:
3(13/5)z - 3y + 6z = 6
Multiply through by 5 to eliminate fractions:
39z - 15y + 30z = 30
Combine like terms:
69z - 15y = 30
15y = 69z - 30
y = (69z - 30)/15
y = (23z - 10)/5
So, the solutions to the system of linear equations are:
x = (13/5)z
y = (23z - 10)/5
z = z
These solutions can be checked algebraically by substituting the values of x, y, and z into the original equations and checking if they satisfy all three equations simultaneously.
To solve the system of linear equations, we will use the method of elimination or substitution. The parameter 'a' is not given in the system of equations you provided, so we can ignore it for now. We will solve for x, y, and z.
First, let's use the elimination method to solve for x, y, and z.
1) Multiply equation (2) by 3:
3(x + 2y - z) = 3(8) -> 3x + 6y - 3z = 24
2) Add equation (1) and equation (3):
(3x - 3y + 6z) + (5x - 8y + 13z) = 6 + 4
8x - 11y + 19z = 10
3) Add the modified equation (2) and equation (3):
(3x + 6y - 3z) + (5x - 8y + 13z) = 24 + 4
8x - 2y + 10z = 28
Now we have the following system of equations:
8x - 11y + 19z = 10
8x - 2y + 10z = 28
To eliminate z, we will subtract the modified equation (2) from equation (1):
(8x - 11y + 19z) - (8x - 2y + 10z) = 10 - 28
-9y + 9z = -18
Dividing both sides of the equation by -9, we get:
y - z = 2 -> equation (4)
Now we have three equations:
8x - 11y + 19z = 10 -> equation (1)
8x - 2y + 10z = 28 -> equation (2)
y - z = 2 -> equation (4)
From equation (4), we can express y in terms of z:
y = z + 2 -> equation (5)
Now, substitute equation (5) into equation (2):
8x - 2(z + 2) + 10z = 28
8x - 2z - 4 + 10z = 28
8x + 8z = 32
x + z = 4 -> equation (6)
We now have two equations:
x + z = 4 -> equation (6)
8x + 8z = 32 -> equation (7)
From equation (6), we can express x in terms of z:
x = 4 - z -> equation (8)
Now substitute equation (8) into equation (7):
8(4 - z) + 8z = 32
32 - 8z + 8z = 32
32 = 32
Since the equation is true, we can see that z can be any real number. Let's choose z = a, where a is a parameter.
Using this value of z = a, we can now find the values of x and y:
From equation (8):
x = 4 - a
From equation (5):
y = a + 2
So the solution to the system of equations, in terms of the parameter a, is:
x = 4 - a
y = a + 2
z = a
To check this solution algebraically, substitute the values of x, y, and z back into the original equations and verify that they satisfy all three equations.
I hope this explanation helps you solve the system of linear equations! If you have any further questions, please, let me know.