document.write( "Question 840152: A contractor buys 14 yd of nylon carpet and 18 yd of wool carpet for $1648. A second purchase, at the same prices, includes 18 yd of nylon carpet and 23 yd of wool carpet for $2111. Find the cost per yard of the wool carpet.
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Algebra.Com's Answer #506030 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( "Let x  and y represent the cost of the nylon and wool carpets respectively
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document.write( "Find the cost \"highlight_green%28y%29\" per yard of the wool carpet.
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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%2814%2Ax%2B18%2Ay=1648%2C18%2Ax%2B23%2Ay=2111%29\"
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document.write( "  First let \"A=%28matrix%282%2C2%2C14%2C18%2C18%2C23%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 \"1648\" and \"2111\" which are highlighted here: 
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document.write( "  \"system%2814%2Ax%2B18%2Ay=highlight%281648%29%2C18%2Ax%2B23%2Ay=highlight%282111%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=%2814%29%2823%29-%2818%29%2818%29=-2\". 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%281648%29%2C18%2Chighlight%282111%29%2C23%29%29\"
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document.write( "  Now compute the determinant of \"A%5Bx%5D\" to get \"abs%28A%5Bx%5D%29=%281648%29%2823%29-%2818%29%282111%29=-94\". 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=%28-94%29%2F%28-2%29=47\"
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document.write( "  So the first solution is \"x=47\"
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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%2C14%2C18%2C18%2C23%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%2C14%2Chighlight%281648%29%2C18%2Chighlight%282111%29%29%29\"
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document.write( "  Now compute the determinant of \"A%5By%5D\" to get \"abs%28A%5By%5D%29=%2814%29%282111%29-%281648%29%2818%29=-110\".
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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=%28-110%29%2F%28-2%29=55\"
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document.write( "  So the second solution is \"y=55\"
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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=47\" and \"y=55\" giving the ordered pair (47, 55)
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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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