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Difficulty: MediumSystems of Linear Equations
Consider the system of equations below.
2(x+3y)−5y=103(x−y)+2x=3\begin{aligned} 2(x + 3y) - 5y &= 10 \\ 3(x - y) + 2x &= 3 \end{aligned}
If (x,y)(x, y) is the solution to the system, what is the value of x+2yx + 2y?
  1. A
    9
  2. B
    10
  3. 11Answer
  4. D
    17

Answer

The value of 11 is the correct answer.
The correct answer is 11. Simplifying the first equation gives 2x+y=102x + y = 10, which allows us to write y=10−2xy = 10 - 2x. The second equation simplifies to 5x−3y=35x - 3y = 3. Substituting the expression for yy into the second equation yields 5x−3(10−2x)=35x - 3(10 - 2x) = 3. Distributing the −3-3 gives 5x−30+6x=35x - 30 + 6x = 3, which simplifies to 11x=3311x = 33, or x=3x = 3. Substituting x=3x = 3 back into the equation for yy gives y=10−2(3)=4y = 10 - 2(3) = 4. Evaluating the expression x+2yx + 2y with these values results in 3+2(4)=113 + 2(4) = 11.

Step-by-Step Solution

1
Simplify both equations in the system by distributing coefficients and combining like terms.
The first equation becomes 2x+6y−5y=10⇒2x+y=102x + 6y - 5y = 10 \Rightarrow 2x + y = 10.
The second equation becomes 3x−3y+2x=3⇒5x−3y=33x - 3y + 2x = 3 \Rightarrow 5x - 3y = 3.
Simplifying the equations makes it easier to use substitution or elimination methods.
2
Solve the simplified system using substitution.
From 2x+y=102x + y = 10, express yy as y=10−2xy = 10 - 2x. Substitute this into the second equation: 5x−3(10−2x)=3⇒5x−30+6x=3⇒11x=33⇒x=35x - 3(10 - 2x) = 3 \Rightarrow 5x - 30 + 6x = 3 \Rightarrow 11x = 33 \Rightarrow x = 3. Substituting x=3x = 3 back into the expression for yy gives y=10−2(3)=4y = 10 - 2(3) = 4.
This determines the unique values of the variables xx and yy that satisfy both equations.
3
Evaluate the expression x+2yx + 2y using the solved values.
Substitute x=3x = 3 and y=4y = 4 into x+2yx + 2y to get 3+2(4)=113 + 2(4) = 11.
This finds the specific quantity requested by the question.

Key Concept

Solving systems of linear equations using algebraic simplification and substitution.
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Topics

Systems of Linear Equations

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