x+y+5z=18
x+y+7z=26
x-4y-7y=-35
its a solve a 3 equation solve the system
To solve the system of equations:
1. Write down the given equations:
1) x + y + 5z = 18
2) x + y + 7z = 26
3) x - 4y - 7z = -35
2. There are several methods to solve this system, such as substitution, elimination, or using matrices. Here, we will use the elimination method.
3. Start by eliminating one variable from two different equations. In this case, the "y" variable is a good choice to eliminate. To do that, subtract equation 1) from equation 2):
(x + y + 7z) - (x + y + 5z) = 26 - 18
y - 5z = 8
4. Now eliminate the "x" variable by subtracting equation 1) from equation 3):
(x - 4y - 7z) - (x + y + 5z) = -35 - 18
-5y - 12z = -53
5. Now we have a system of two equations with two variables:
1) y - 5z = 8
2) -5y - 12z = -53
6. We can solve this system of equations using the elimination or substitution method again. To make the process easier, let's multiply equation 1) by 5 and equation 2) by 1:
5(y - 5z) = 5(8)
-25y - 60z = -265
-5y - 12z = -53
7. Now, add the two equations together:
(-25y - 60z) + (-5y - 12z) = -265 + (-53)
-30y - 72z = -318
8. Simplify the equation:
-30y - 72z = -318
9. Now we have two equations with one variable each:
1) -5y - 12z = -53
2) -30y - 72z = -318
10. Solve either equation 1) or 2) for one variable (preferably "y" to avoid fractions):
-5y = -53 + 12z
-5y = -53 + 12z
y = (53 - 12z) / 5
11. Substitute this value of "y" back into any of the two equations and solve for "z". Let's substitute it into equation 1):
-5((53 - 12z) / 5) - 12z = -53
-53 + 12z - 12z = -53
-53 = -53
The equation simplifies to 0 = 0, which means "z" can be any value since it cancels out.
12. Now, substitute the value of "z" back into equation 1) to solve for "y":
y - 5z = 8
y - 5(0) = 8
y = 8
So, "y" is equal to 8.
13. Finally, substitute the values of "y" and "z" back into any of the three original equations to solve for "x". Let's use equation 1):
x + y + 5z = 18
x + 8 + 5(0) = 18
x + 8 = 18
x = 10
Therefore, the solution to the system of equations is x = 10, y = 8, and z can be any value.