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Difficulty: MediumSystems of Linear Equations
x−2y=53x−y=10\begin{aligned} x - 2y &= 5 \\ 3x - y &= 10 \end{aligned}

If (x,y)(x, y) is the solution to the system of equations above, what is the value of x+yx + y?

  1. 2Answer
  2. B
    3
  3. C
    6
  4. D
    8

Answer

2
The correct answer is 2. Isolating xx in the first equation gives x=2y+5x = 2y + 5. Substituting this expression into the second equation yields 3(2y+5)−y=103(2y + 5) - y = 10. Distributing the 3 gives 6y+15−y=106y + 15 - y = 10, which simplifies to 5y+15=105y + 15 = 10. Subtracting 15 from both sides gives 5y=−55y = -5, so y=−1y = -1. Substituting y=−1y = -1 back into the equation for xx gives x=2(−1)+5=3x = 2(-1) + 5 = 3. The sum of the solutions is x+y=3+(−1)=2x + y = 3 + (-1) = 2.

Step-by-Step Solution

1
Isolate xx in the first equation.
x=2y+5x = 2y + 5
Isolating one variable makes it straightforward to substitute it into the other equation.
2
Substitute the expression for xx into the second equation and solve for yy.
y=−1y = -1
Substituting x=2y+5x = 2y + 5 gives 3(2y+5)−y=103(2y + 5) - y = 10. Distributing the 3 yields 6y+15−y=106y + 15 - y = 10. Combining like terms gives 5y+15=105y + 15 = 10. Subtracting 15 from both sides gives 5y=−55y = -5, so y=−1y = -1.
3
Substitute y=−1y = -1 back into the expression for xx to find its value.
x=3x = 3
Using the isolated expression x=2y+5x = 2y + 5, substituting y=−1y = -1 gives x=2(−1)+5=−2+5=3x = 2(-1) + 5 = -2 + 5 = 3.
4
Calculate the sum x+yx + y.
2
The question asks for the value of x+yx + y, which is 3+(−1)=23 + (-1) = 2.

Key Concept

Solving systems of linear equations in two variables using substitution or elimination, and evaluating algebraic combinations of the variables.

Alternative Method

The system can also be solved by elimination. Multiplying the second equation by 2 gives 6x−2y=206x - 2y = 20. Subtracting the first equation (x−2y=5x - 2y = 5) from this new equation eliminates yy, resulting in 5x=155x = 15, or x=3x = 3. Substituting x=3x = 3 into either original equation yields y=−1y = -1. Finally, the sum is x+y=3+(−1)=2x + y = 3 + (-1) = 2.
Estimated Time:1m 30s
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Topics

Systems of Linear Equations

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