document.write( "Question 432065: Solve each system of equations using Cramer's Rule if it is applicable. If Cramer's Rule is not applicable, say so.\r
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Algebra.Com's Answer #299694 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( "Solve system of equations using Cramer's Rule
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document.write( "x+2y=5
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document.write( "x-y=3
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document.write( " x = 11/3 and  y = 2/3  See below
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Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 2 variables

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document.write( "  \"system%281%2Ax%2B2%2Ay=5%2C1%2Ax%2B-1%2Ay=3%29\"
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document.write( "  First let \"A=%28matrix%282%2C2%2C1%2C2%2C1%2C-1%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 \"5\" and \"3\" which are highlighted here: 
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document.write( "  \"system%281%2Ax%2B2%2Ay=highlight%285%29%2C1%2Ax%2B-1%2Ay=highlight%283%29%29\"
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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=%281%29%28-1%29-%282%29%281%29=-3\". Remember that the determinant of the 2x2 matrix \"A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29\" is \"abs%28A%29=ad-bc\". If you need help with calculating the determinant of any two by two matrices, then 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( "  \"A%5Bx%5D=%28matrix%282%2C2%2Chighlight%285%29%2C2%2Chighlight%283%29%2C-1%29%29\"
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document.write( "  Now compute the determinant of \"A%5Bx%5D\" to get \"abs%28A%5Bx%5D%29=%285%29%28-1%29-%282%29%283%29=-11\". Once again, remember that the determinant of the 2x2 matrix \"A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29\" is \"abs%28A%29=ad-bc\"
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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=%28-11%29%2F%28-3%29=11%2F3\"
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document.write( "  So the first solution is \"x=11%2F3\"
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document.write( "  We'll follow the same basic idea to find the other solution. Let's reset by letting \"A=%28matrix%282%2C2%2C1%2C2%2C1%2C-1%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( "  \"A%5Bx%5D=%28matrix%282%2C2%2C1%2Chighlight%285%29%2C1%2Chighlight%283%29%29%29\"
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document.write( "  Now compute the determinant of \"A%5By%5D\" to get \"abs%28A%5By%5D%29=%281%29%283%29-%285%29%281%29=-2\".
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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=%28-2%29%2F%28-3%29=2%2F3\"
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document.write( "  So the second solution is \"y=2%2F3\"
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document.write( "  ====================================================================================
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document.write( "  Final Answer:
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document.write( "  So the solutions are \"x=11%2F3\" and \"y=2%2F3\" giving the ordered pair (11/3, 2/3)
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document.write( "  Once again, Cramer's Rule is dependent on determinants. Take a look at this 2x2 Determinant Solver if you need more practice with determinants.
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