Hey! I'm getting stuck on how to solve this system of linear equations. Can anyone show me step by step Thanks!
3x-5y+5=1
5x-2y+3z=0
7x-y+3z=0
3x-5y+5=1
5x-2y+3z=0
7x-y+3z=0
Do you have a typo?
3x - 5y + 5? = 1
There is no z term in equation 1?
yes that was a typo, my bad!
Its supposed to be 3x-5y+5z=1
Sure! To solve this system of linear equations, we can use the method of elimination or substitution. Let's go with elimination.
Step 1: Equating coefficients
First, we need to make the coefficients of one of the variables the same in at least two equations. Looking at the given equations, we see that the variable "y" has a coefficient of -5 in the first equation and -1 in the second equation. To eliminate "y", we need to multiply the first equation by -1 and the second equation by -5. This will make the coefficients of "y" the same in both equations.
The equations become:
-3x + 5y - 5 = -1 (equation 1 multiplied by -1)
-25x + 10y - 15z = 0 (equation 2 multiplied by -5)
7x - y + 3z = 0 (unchanged equation)
Step 2: Add the equations
Now, we add these equations together, term by term, to eliminate "y".
(-3x + 5y - 5) + (-25x + 10y - 15z) + (7x - y + 3z) = 0
Simplifying the above equation gives us:
-21x - 12z = -6 (equation 3)
Step 3: Continue eliminating variables
We have now eliminated "y". Next, we can eliminate "z" by multiplying the equation 2 by 3 and adding it to equation 3.
-75x + 30y - 45z = 0 (equation 2 multiplied by 3)
(-75x + 30y - 45z) + (-21x - 12z) = -6
Simplifying the above equation gives us:
-96x + 18y = -6 (equation 4)
Step 4: Solve for a single variable
Now we have a system of two equations:
-21x - 12z = -6 (equation 3)
-96x + 18y = -6 (equation 4)
From equation 4, we can solve for "y":
18y = -6 + 96x
y = (-6 + 96x) / 18
y = (-1 + 16x) / 3 (equation 5)
Step 5: Substitute variables back
Now that we have an expression for "y" in terms of "x", we can substitute it back into the original equations to find the values of "x" and "z".
Substituting equation 5 into equation 3, we get:
-21x - 12z = -6
Substituting equation 5 into equation 4, we get:
-96x + 18((-1 + 16x) / 3) = -6
Step 6: Simplify and solve
Now, we simplify and solve the remaining equations.
-21x - 12z = -6 (equation 6)
-96x - 6 + 288x = -6 (equation 7)
Simplifying equation 7 gives us:
192x = -6 + 6
192x = 0
x = 0
Substituting the value of x into equation 6 gives us:
-21(0) - 12z = -6
-12z = -6
z = -6 / -12
z = 0.5
Finally, we have x = 0, y = (-1 + 16x) / 3 = -1 / 3, and z = 0.5.
Therefore, the solution to the system of linear equations is x = 0, y = -1/3, and z = 0.5.