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Difficulty: MediumSystems of Linear Equations

In the system of equations below, xx and yy are variables.

2x−y=52x - y = 5
3x−2y=−173x - 2y = -17

What is the value of xx?

  1. A
    7
  2. B
    25
  3. 27Answer
  4. D
    49

Answer

The value of xx is 27.
Isolating yy from the first equation gives y=2x−5y = 2x - 5. Substituting this expression into the second equation yields 3x−2(2x−5)=−173x - 2(2x - 5) = -17. Distributing the −2-2 gives 3x−4x+10=−173x - 4x + 10 = -17, which simplifies to −x+10=−17-x + 10 = -17. Subtracting 1010 from both sides results in −x=−27-x = -27, meaning x=27x = 27. Alternatively, multiplying the first equation by 22 gives 4x−2y=104x - 2y = 10, and subtracting the second equation from it eliminates yy directly to give x=10−(−17)=27x = 10 - (-17) = 27.

Step-by-Step Solution

1
Isolate the variable yy in the first equation, 2x−y=52x - y = 5.
y=2x−5y = 2x - 5
This allows for substitution of the expression for yy into the second equation.
2
Substitute the expression for yy into the second equation, 3x−2y=−173x - 2y = -17.
3x−2(2x−5)=−173x - 2(2x - 5) = -17
To create a single-variable equation in terms of xx.
3
Distribute the coefficient −2-2 to the terms inside the parentheses and simplify the equation.
3x−4x+10=−173x - 4x + 10 = -17, which simplifies to −x+10=−17-x + 10 = -17
Applying the distributive property correctly, noting that −2×−5=10-2 \times -5 = 10.
4
Subtract 1010 from both sides of the equation to isolate the term with xx.
−x=−27-x = -27
To isolate the variable term on one side of the equation.
5
Multiply both sides of the equation by −1-1 to solve for xx.
x=27x = 27
To find the positive value of the variable xx.

Key Concept

Solving systems of linear equations using the substitution method.
Estimated Time:1m 30s
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Topics

Systems of Linear Equations

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