I NEED help on the following problem can some one help me please and thanks
5x-y+z= 8
2x+2y-3z= 28
x-3y+2z= -15
It is difficult to assist you without knowing how you are supposed to solve it. I would use row reduction, but you may be at some other learning level.
Certainly! I'd be happy to help you with this linear system of equations.
To solve this system, we can use a common method called the elimination method. Here's how you can proceed:
Step 1: Choose a pair of equations and eliminate one variable.
Let's choose the first and second equations. We can eliminate the variable "y" by multiplying the first equation by 2 and the second equation by -1, making the coefficients of "y" in both equations the same:
2*(5x - y + z) = 2*8
-1*(2x + 2y - 3z) = -1*28
Simplifying these equations, we get:
10x - 2y + 2z = 16
-2x - 2y + 3z = -28
Step 2: Add the modified equations together.
By adding the modified equations, we can eliminate the "y" variable:
(10x - 2y + 2z) + (-2x - 2y + 3z) = 16 + (-28)
This simplifies to:
8x + 5z = -12
Step 3: Now, choose a different pair of equations and eliminate the same variable ("y" in this case).
Let's choose the first and third equations. We can eliminate "y" by multiplying the first equation by 3 and the third equation by 1, making the coefficients of "y" in both equations the same:
3*(5x - y + z) = 3*8
1*(x - 3y + 2z) = 1*(-15)
Simplifying these equations, we get:
15x - 3y + 3z = 24
x - 3y + 2z = -15
Step 4: Add the modified equations together.
By adding the modified equations, we can eliminate the "y" variable:
(15x - 3y + 3z) + (x - 3y + 2z) = 24 + (-15)
This simplifies to:
16x + 5z = 9
Now, we have two equations:
8x + 5z = -12 (from Step 2)
16x + 5z = 9 (from Step 4)
Step 5: Solve for one variable.
We can choose either x or z to solve for. Let's solve for x using the first equation:
8x = -12 - 5z
x = (-12 - 5z) / 8
Step 6: Substitute the value of x in the second equation.
Substituting the value of x in the second equation, we get:
16((-12 - 5z) / 8) + 5z = 9
Now, we have an equation in terms of z that we can solve.
Step 7: Solve for z.
Simplifying the equation, we get:
(-24 - 10z) + 5z = 9
-24 - 5z = 9
-5z = 9 + 24
-5z = 33
z = 33 / -5
z = -6.6
Step 8: Substitute the value of z in x = (-12 - 5z) / 8
Substituting the value of z in the equation for x, we get:
x = (-12 - 5(-6.6)) / 8
x = (-12 + 33) / 8
x = 21 / 8
x = 2.625
Step 9: Calculate the value of y.
To find y, we can substitute the values of x and z in one of the original equations. Let's use the first equation:
5x - y + z = 8
5(2.625) - y + (-6.6) = 8
13.125 - y - 6.6 = 8
- y = 8 - 13.125 + 6.6
- y = 1.475
y = -1.475
Therefore, the solution to the given system of equations is:
x = 2.625, y = -1.475, z = -6.6
I hope this explanation helps you solve the problem! Let me know if you have any further questions.