document.write( "Question 899057: what is the value of a,b,c\r
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Algebra.Com's Answer #545148 by ewatrrr(24785)\"\" \"About 
You can put this solution on YOUR website!
 
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document.write( "Hi
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document.write( "2a-5b+c=1, c = 2-a  and b = -3 + 3c
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document.write( "2a - 5(-3+3(2-a)) + (2-a)= 1
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document.write( "2a - 5(3 - 3a) + 2 - a = 1
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document.write( "2a - 15 + 15a + 2 - a = 1
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document.write( "  16a = 14
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document.write( "    a = 14/16 = 7/8 
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document.write( "    c = \"16%2F8+-+7%2F8\" = 9/8
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document.write( "    b = \"-3+%2B+27%2F8+=+-24%2F8+%2B+27%2F8\" = 3/8
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Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 3 variables

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document.write( "  \"system%282%2Ax%2B-5%2Ay%2B1%2Az=1%2C1%2Ax%2B0%2Ay%2B1%2Az=2%2C0%2Ax%2B1%2Ay%2B-3%2Az=-3%29\"
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document.write( "  First let \"A=%28matrix%283%2C3%2C2%2C-5%2C1%2C1%2C0%2C1%2C0%2C1%2C-3%29%29\". This is the matrix formed by the coefficients of the given system of equations.
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document.write( "  Take note that the right hand values of the system are \"1\", \"2\", and \"-3\" and they are highlighted here: 
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document.write( "  These values are important as they will be used to replace the columns of the matrix A.
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document.write( "  Now let's calculate the the determinant of the matrix A to get \"abs%28A%29=-16\". To save space, I'm not showing the calculations for the determinant. However, if you need help with calculating the determinant of the matrix A, check out this solver.
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document.write( "  Notation note: \"abs%28A%29\" denotes the determinant of the matrix A.
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document.write( "  Now replace the first column of A (that corresponds to the variable 'x') with the values that form the right hand side of the system of equations. We will denote this new matrix \"A%5Bx%5D\" (since we're replacing the 'x' column so to speak).
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document.write( "  Now compute the determinant of \"A%5Bx%5D\" to get \"abs%28A%5Bx%5D%29=-14\". Again, as a space saver, I didn't include the calculations of the determinant. Check out this solver to see how to find this determinant.
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document.write( "  To find the first solution, simply divide the determinant of \"A%5Bx%5D\" by the determinant of \"A\" to get: \"x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%28-14%29%2F%28-16%29=7%2F8\"
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document.write( "  So the first solution is \"x=7%2F8\"
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document.write( "  We'll follow the same basic idea to find the other two solutions. Let's reset by letting \"A=%28matrix%283%2C3%2C2%2C-5%2C1%2C1%2C0%2C1%2C0%2C1%2C-3%29%29\" again (this is the coefficient matrix).
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document.write( "  Now replace the second column of A (that corresponds to the variable 'y') with the values that form the right hand side of the system of equations. We will denote this new matrix \"A%5By%5D\" (since we're replacing the 'y' column in a way).
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document.write( "  Now compute the determinant of \"A%5By%5D\" to get \"abs%28A%5By%5D%29=-6\".
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document.write( "  To find the second solution, divide the determinant of \"A%5By%5D\" by the determinant of \"A\" to get: \"y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%28-6%29%2F%28-16%29=3%2F8\"
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document.write( "  So the second solution is \"y=3%2F8\"
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document.write( "  Let's reset again by letting \"A=%28matrix%283%2C3%2C2%2C-5%2C1%2C1%2C0%2C1%2C0%2C1%2C-3%29%29\" which is the coefficient matrix.
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document.write( "  Replace the third column of A (that corresponds to the variable 'z') with the values that form the right hand side of the system of equations. We will denote this new matrix \"A%5Bz%5D\" 
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document.write( "  Now compute the determinant of \"A%5Bz%5D\" to get \"abs%28A%5Bz%5D%29=-18\".
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document.write( "  To find the third solution, divide the determinant of \"A%5Bz%5D\" by the determinant of \"A\" to get: \"z=%28abs%28A%5Bz%5D%29%29%2F%28abs%28A%29%29=%28-18%29%2F%28-16%29=9%2F8\"
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document.write( "  So the third solution is \"z=9%2F8\"
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document.write( "  ====================================================================================
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document.write( "  Final Answer:
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document.write( "  So the three solutions are \"x=7%2F8\", \"y=3%2F8\", and \"z=9%2F8\" giving the ordered triple (7/8, 3/8, 9/8)
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document.write( "  Note: there is a lot of work that is hidden in finding the determinants. Take a look at this 3x3 Determinant Solver to see how to get each determinant.
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