Question 473147: im lost please help me with this system:
4a+7b=-39
8a-4c=-24
6b-4c=-46
Found 3 solutions by stanbon, ewatrrr, karaoz:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! 4a+7b+0c=-39
8a+0b-4c=-24
0a+6b-4c=-46
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I used a matrix method to get:
a = -1
b = -5
c = 4
============
Cheers,
Stan H.
Answer by ewatrrr(24785)  (Show Source):
You can put this solution on YOUR website!
Hi,
Taking it a 'step' st a time:
4a+7b=-39
8a-4c=-24
6b-4c=-46
8a-4c=-24
-6b+4c=+46
8a-6b = 22 a = (22-6b)/8 = (11-3b)/4
4a+7b=-39
4(11-3b)/4 +7b=-39
11-3b +7b=-39
4b = -50
b = -12.5 and a = 97/8 = 12.125 a = (22-6(-12.5))/8
CHECKING our Answer***
4a+7b=-39
4*12.125 - 87.5 = 48.5 -87.5 = 39
Answer by karaoz(32)  (Show Source):
You can put this solution on YOUR website!
Before you get confident in solving these types of questions, it is a good idea to first re-write your system in standard form.
This basically means that you need to reserve one and the same column for each single variable on the left hand side and keep any constant on the right-hand side.
In your example, one possible standard form looks like this:
4a + 7b = -39
8a - 4c = -24
6b - 4c = -46
While there are many other possible standard forms, any one will serve the purpose. So, we will work on this one.
First step is to reduce the system, which is currently 3 equations with 3 unknowns, down to 2 equations with 2 unknowns.
This can be achieved by choosing one of the three variables, which we will want to eliminate.
The question is: Which one?
Theoretically, it does not matter which one.
From practical perspective, however, you will want to eliminate the one which will create the least amount of work.
In this example, variable c can be eliminated by subtracting third equation from the second equation:
(8a - 4c = -24)
- ( 6b - 4c = -46)
----------------------
8a - 6b = 22
Now we have new equation that does not have c in it and we already had one without c in it (1st equation).
These two equations are now a reduced system of two equations with two unknowns.
First step is done.
Second step is to further reduce the system from two equations with 2 unknowns down to 1 equation with 1 unknown.
Currently, the system is:
4a + 7b = -39
8a - 6b = 22
and for this one, it will be easier to eliminate variable a rather than b.
To eliminate a, we will first multiply the first equation by (-2) and then add the two equations.
(4a + 7b = -39) * (-2)
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-8a - 14b = 78
Now, we add the two equations:
-8a - 14b = 78
8a - 6b = 22
----------------------
- 20b = 100,
which is one equation with one unknown.
Step 2 is done.
Step 3 is to solve this simple equation.
The solution is: b = -5.
Step 3 is done.
Step 4 is to use the value obtained for one variable and substitute this value into any equation obtained in step 2 that also contains the other variable and solve for that variable.
We can take equation
8a - 6b = 22
and substitute b = -5 to get
8a - 6(-5) = 22
Solve for a:
8a + 30 = 22
8a = -8
a = -1
Step 4 is done.
Step 5 is to use the known values of two variables and substitute into any of the three original equations that also contain the remaining unknown variable.
Clearly, we cannot take the first equation since that one does not contain the remaining unknown variable c.
So, we can take either the second or the third.
Let's take the second.
8a - 4c = -24
Substitute a = -1 to get:
8(-1) - 4c = -24
Solve for c.
-8 - 4c = -24
-4c = -16
c = 4.
Step 5 is done.
We now solved the system: a = -1, b = -5, c = 4.
The last step would be to check the solution by substituting these values into the original three equations and see if they all give us true statements.
4(-1) + 7(-5) = -39
8(-1) - 4(4) = -24
6(-5) - 4(4) = -46
Simplifying the left hand sides we have:
-39 = -39
-24 = -24
-46 = -46,
which are all true statements.
Therefore our solution is correct.
a = -1, b = -5, c = 4
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