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Difficulty: MediumLinear Equations in One and Two Variables
If real numbers xx and yy satisfy the system of equations:
3x−5y=283x - 5y = 28
x(x−2y)=3y(x−2y)x(x - 2y) = 3y(x - 2y)
where x≠2yx \neq 2y, what is the value of x+yx + y?
  1. A
    7
  2. B
    21
  3. 28Answer
  4. D
    35
  5. E
    84

Answer

The value of x+yx + y is 28.
Because x≠2yx \neq 2y, the expression (x−2y)(x - 2y) is non-zero. Dividing both sides of the second equation x(x−2y)=3y(x−2y)x(x - 2y) = 3y(x - 2y) by (x−2y)(x - 2y) gives x=3yx = 3y. Substituting x=3yx = 3y into the first equation 3x−5y=283x - 5y = 28 yields 3(3y)−5y=283(3y) - 5y = 28, which simplifies to 4y=284y = 28, so y=7y = 7. Then x=3(7)=21x = 3(7) = 21. The required sum x+yx + y is 21+7=2821 + 7 = 28.

Step-by-Step Solution

1
Simplify the second equation using the given constraint.
Since x≠2yx \neq 2y, the factor (x−2y)≠0(x - 2y) \neq 0. Dividing both sides of x(x−2y)=3y(x−2y)x(x - 2y) = 3y(x - 2y) by (x−2y)(x - 2y) yields x=3yx = 3y.
Dividing by a non-zero expression is algebraically valid and simplifies the second equation into a linear relationship.
2
Substitute x=3yx = 3y into the first linear equation.
3(3y)−5y=28  ⟹  9y−5y=28  ⟹  4y=28  ⟹  y=73(3y) - 5y = 28 \implies 9y - 5y = 28 \implies 4y = 28 \implies y = 7.
Substitution reduces the system to a single linear equation in one variable.
3
Calculate xx and the required sum x+yx + y.
x=3(7)=21x = 3(7) = 21, so x+y=21+7=28x + y = 21 + 7 = 28.
Demonstrates full solution of the target expression.

Key Concept

Solving Systems of Linear Equations by Algebraic Reduction
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Topics

Linear Equations in One and Two Variables

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