{4x-2y+5z=6 {3x+3y+8z=4 {x-5y-3z=5
1 answer:
There are three possible outcomes that you may encounter when working with these system of equations:
- one solution
- no solution
- infinite solutions
We are going to try and find values of x, y, and z that will satisfy all three equations at the same time. The following are the equations:
We are going to use elimination(or addition) method
Step 1: Choose to eliminate any one of the variables from any pair of equations.
In this case it looks like if we multiply the third equation by 4 and subtracting it from equation 1, it will be fairly simple to eliminate the x term from the first and third equation.
So multiplying Left Hand Side(L.H.S) and Right Hand Side(R.H.S) of 3rd equation with 4 gives us a new equation 4.:
4. 4x-20y-12z = 20
Subtracting eq. 4 from Eq. 1:
(L.HS) : 4x-2y+5z-(4x-20y-12z) = 18y+17z
(R.H.S) : 20 - 6 = 14
5. 18y+17z=14
Step 2: Eliminate the SAME variable chosen in step 2 from any other pair of equations, creating a system of two equations and 2 unknowns.
Similarly if we multiply 3rd equation with 3 and then subtract it from eq. 2 we get:
(L.HS) : 3x+3y+8z-(3x-15y-9z) = 18y+17z
(R.H.S) : 4 - 15 = -11
6. 18y+17z = -11
Step 3: Solve the remaining system of equations 6 and 5 found in step 2 and 1.
Now if we try to solve equations 5 and 6 for the variables y and z. Subtracting eq 6 from eq. 5 we get:
(L.HS) : 18y+17z-(18y+17z) = 0
(R.HS) : 14-(-11) = 25
0 = 25
which is false, hence no solution exists
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Answers:
Lower bound = 41.3
Upper bound = 41.5
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Explanation:
49.15 is the smallest f can be while 49.24 is the largest it can be. Well technically we could have something like 49.2499 or 49.24999 and so on. We slowly approach 49.25 but never actually get there
So the variable f is between 49.15 and 49.25 inclusive of the first value but excluding the second value. We can write it as
For similar reasoning,
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If we wanted to subtract those variables and get the smallest result possible, then we need to pick values that are closest together. It might help to set up a number line.
This means we'd go for f = 49.15 and g = 7.85
The lower bound for f-g is f-g = 49.15-7.85 = 41.3
In contrast, the upper bound is when the two variables are spaced as far apart as possible. The upper bound is f-g = 49.25 - 7.75 = 41.5