document.write( "Question 473147: im lost please help me with this system:\r
\n" ); document.write( "\n" ); document.write( "4a+7b=-39
\n" ); document.write( "8a-4c=-24
\n" ); document.write( "6b-4c=-46 \n" ); document.write( "

Algebra.Com's Answer #324450 by karaoz(32) $\"\"$ $\"About$

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.
\n" ); document.write( "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.
\n" ); document.write( "
In your example, one possible standard form looks like this:
\n" ); document.write( "

\r\n" );
document.write( "4a + 7b      = -39\r\n" );
document.write( "8a      - 4c = -24\r\n" );
document.write( "     6b - 4c = -46\r\n" );
document.write( "

While there are many other possible standard forms, any one will serve the purpose. So, we will work on this one.
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First step is to reduce the system, which is currently 3 equations with 3 unknowns, down to 2 equations with 2 unknowns.
\n" ); document.write( "This can be achieved by choosing one of the three variables, which we will want to eliminate.
\n" ); document.write( "The question is: Which one?
\n" ); document.write( "Theoretically, it does not matter which one.
\n" ); document.write( "From practical perspective, however, you will want to eliminate the one which will create the least amount of work.
\n" ); document.write( "In this example, variable c can be eliminated by subtracting third equation from the second equation:
\n" ); document.write( "

\r\n" );
document.write( "  (8a      - 4c = -24)\r\n" );
document.write( "- (     6b - 4c = -46)\r\n" );
document.write( "----------------------\r\n" );
document.write( "   8a - 6b      =  22\r\n" );
document.write( "

Now we have new equation that does not have c in it and we already had one without c in it (1st equation).
\n" ); document.write( "These two equations are now a reduced system of two equations with two unknowns.
\n" ); document.write( "First step is done.

\n" ); document.write( "Second step is to further reduce the system from two equations with 2 unknowns down to 1 equation with 1 unknown.
\n" ); document.write( "Currently, the system is:
\n" ); document.write( "

\r\n" );
document.write( "4a + 7b = -39\r\n" );
document.write( "8a - 6b =  22\r\n" );
document.write( "

and for this one, it will be easier to eliminate variable a rather than b.
\n" ); document.write( "To eliminate a, we will first multiply the first equation by (-2) and then add the two equations.

\r\n" );
document.write( "(4a +  7b = -39) * (-2)\r\n" );
document.write( "----------------------\r\n" );
document.write( "-8a - 14b =  78\r\n" );
document.write( "

Now, we add the two equations:

\r\n" );
document.write( "-8a - 14b =  78\r\n" );
document.write( " 8a -  6b =  22\r\n" );
document.write( "----------------------\r\n" );
document.write( "    - 20b = 100,\r\n" );
document.write( "

which is one equation with one unknown.
\n" ); document.write( "Step 2 is done.
\n" ); document.write( "
Step 3 is to solve this simple equation.
\n" ); document.write( "The solution is: b = -5.
\n" ); document.write( "Step 3 is done.
\n" ); document.write( "
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.
\n" ); document.write( "We can take equation
\n" ); document.write( "8a - 6b = 22
\n" ); document.write( "and substitute b = -5 to get
\n" ); document.write( "8a - 6(-5) = 22
\n" ); document.write( "Solve for a:
\n" ); document.write( "8a + 30 = 22
\n" ); document.write( "8a = -8
\n" ); document.write( "a = -1
\n" ); document.write( "Step 4 is done.
\n" ); document.write( "
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.
\n" ); document.write( "Clearly, we cannot take the first equation since that one does not contain the remaining unknown variable c.
\n" ); document.write( "So, we can take either the second or the third.
\n" ); document.write( "Let's take the second.
\n" ); document.write( "8a - 4c = -24
\n" ); document.write( "Substitute a = -1 to get:
\n" ); document.write( "8(-1) - 4c = -24
\n" ); document.write( "Solve for c.
\n" ); document.write( "-8 - 4c = -24
\n" ); document.write( "-4c = -16
\n" ); document.write( "c = 4.
\n" ); document.write( "Step 5 is done.
\n" ); document.write( "
We now solved the system: a = -1, b = -5, c = 4.
\n" ); document.write( "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.
\n" ); document.write( "

\r\n" );
document.write( "4(-1) + 7(-5)        = -39\r\n" );
document.write( "8(-1)         - 4(4) = -24\r\n" );
document.write( "        6(-5) - 4(4) = -46\r\n" );
document.write( "

Simplifying the left hand sides we have:
\n" ); document.write( "-39 = -39
\n" ); document.write( "-24 = -24
\n" ); document.write( "-46 = -46,
\n" ); document.write( "which are all true statements.
\n" ); document.write( "Therefore our solution is correct.

\n" ); document.write( "a = -1, b = -5, c = 4
\n" ); document.write( " \n" ); document.write( "

\n" );