document.write( "Question 278155: hello
\n" ); document.write( "how would i solve this linear system using the cramers rule?\r
\n" ); document.write( "\n" ); document.write( "4x-7y-7z= -26
\n" ); document.write( "3x-7y-9z=-26
\n" ); document.write( "-9x-5y-2z= -51\r
\n" ); document.write( "\n" ); document.write( "this is what i have done but it seems wrong
\n" ); document.write( "i get that d = -145
\n" ); document.write( " dx = -1939/ -145
\n" ); document.write( " dy = 1175/-145
\n" ); document.write( " dz = -1428/-145\r
\n" ); document.write( "\n" ); document.write( "this answers seem to work only with the first equation.\r
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Algebra.Com's Answer #202404 by jim_thompson5910(35256)\"\" \"About 
You can put this solution on YOUR website!
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Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 3 variables

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\n" ); document.write( " First let \"A=%28matrix%283%2C3%2C4%2C-7%2C-7%2C3%2C-7%2C-9%2C-9%2C-5%2C-2%29%29\". This is the matrix formed by the coefficients of the given system of equations.
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\n" ); document.write( " Take note that the right hand values of the system are \"-26\", \"-26\", and \"-51\" and they are highlighted here:
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\n" ); document.write( " These values are important as they will be used to replace the columns of the matrix A.
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\n" ); document.write( " Now let's calculate the the determinant of the matrix A to get \"abs%28A%29=-187\". 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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\n" ); document.write( " Notation note: \"abs%28A%29\" denotes the determinant of the matrix A.
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\n" ); 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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\n" ); document.write( " Now compute the determinant of \"A%5Bx%5D\" to get \"abs%28A%5Bx%5D%29=-454\". 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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\n" ); 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=%28-454%29%2F%28-187%29=454%2F187\"
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\n" ); document.write( " So the first solution is \"x=454%2F187\"
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\n" ); document.write( " We'll follow the same basic idea to find the other two solutions. Let's reset by letting \"A=%28matrix%283%2C3%2C4%2C-7%2C-7%2C3%2C-7%2C-9%2C-9%2C-5%2C-2%29%29\" again (this is the coefficient matrix).
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\n" ); 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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\n" ); document.write( " Now compute the determinant of \"A%5By%5D\" to get \"abs%28A%5By%5D%29=-1181\".
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\n" ); 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=%28-1181%29%2F%28-187%29=1181%2F187\"
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\n" ); document.write( " So the second solution is \"y=1181%2F187\"
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\n" ); document.write( " Let's reset again by letting \"A=%28matrix%283%2C3%2C4%2C-7%2C-7%2C3%2C-7%2C-9%2C-9%2C-5%2C-2%29%29\" which is the coefficient matrix.
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\n" ); 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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\n" ); document.write( " Now compute the determinant of \"A%5Bz%5D\" to get \"abs%28A%5Bz%5D%29=227\".
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\n" ); 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=%28227%29%2F%28-187%29=-227%2F187\"
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\n" ); document.write( " So the third solution is \"z=-227%2F187\"
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\n" ); document.write( " Final Answer:
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\n" ); document.write( " So the three solutions are \"x=454%2F187\", \"y=1181%2F187\", and \"z=-227%2F187\" giving the ordered triple (454/187, 1181/187, -227/187)
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\n" ); 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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