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Question 10 of 18

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Heart of Algebra


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table attributes columnalign right center left columnspacing 2px end attributes row cell 4 x minus y end cell equals cell 3 y plus 7 end cell row cell x plus 8 y end cell equals 4 end table

Based on the system of equations above, what is the value of the product xy?

Select an Answer

minus 3 over 2

Correct Answer: 

1 fourth

Correct Answer: 

1 half

Correct Answer: 

11 over 9

Correct Answer: 

Choice C is correct. There are several solution methods possible, but all involve persevering in solving for the two variables and calculating the product. For example, combining like terms in the first equation yields 4 x space minus space 4 y space equals space 7 and then multiplying that by 2 gives 8 x space minus space 8 y space equals space 14. When this transformed equation is added to the second given equation, the y-terms are eliminated, leaving an equation in just one variable: 9 x equals 18 comma or x equals 2. Substituting 2 for x in the second equation (one could use either to solve) yields 2 plus 8 y equals 4 comma which gives y equals 1 fourth. Finally, the product xy is 2 cross times 1 fourth equals 1 half.

Choice A is not the correct answer. Students who select this option have most likely made a calculation error in transforming the second equation (using minus 4 x minus 8 y equals minus 16 instead of minus 4 x minus 32 y equals minus 16) and used it to eliminate the x-terms.

Choice B is not the correct answer. This is the value of y for the solution of the system, but it has not been put back into the system to solve for x to determine the product xy

Choice D is not the correct answer. Not understanding how to eliminate a variable when solving a system, a student may have added the equations 4 x minus 4 y equals 7 and x plus 8 y equals 4 to yield 5 x space plus space 4 y space equals space 11. From here, a student may mistakenly simplify the left-hand side of this resulting equation to yield 9 x y equals 11 and then proceed to use division by 9 on both sides in order to solve for xy.

Question Difficulty: 

Students must solve a system of linear equations.