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Difficulty: MediumSolving Systems of Linear Equations

Calculate the solution (x,y)(x, y) for the system of linear equations shown below.

3x−2y=73x - 2y = 7
2x+4y=102x + 4y = 10

What is the value of the expression x−yx - y?

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

Answer

The correct value of the expression is 2.
The system can be efficiently solved using either substitution or elimination. By dividing the second equation by 2, we get x+2y=5x + 2y = 5, which means x=5−2yx = 5 - 2y. Substituting this into the first equation gives 3(5−2y)−2y=73(5 - 2y) - 2y = 7. Distributing the 3 results in 15−6y−2y=715 - 6y - 2y = 7, which simplifies to 15−8y=715 - 8y = 7. Subtracting 15 from both sides gives −8y=−8-8y = -8, so y=1y = 1. Substituting y=1y = 1 back into x=5−2yx = 5 - 2y yields x=5−2(1)=3x = 5 - 2(1) = 3. Finally, the requested expression is x−yx - y, which evaluates to 3−1=23 - 1 = 2.

Step-by-Step Solution

1
Simplify the second equation by dividing all terms by 2.
x+2y=5x + 2y = 5
Dividing the entire equation by a common factor makes the coefficients smaller and allows us to easily isolate xx for the substitution method.
2
Isolate xx in the simplified equation.
x=5−2yx = 5 - 2y
Expressing xx in terms of yy prepares us to substitute this expression into the first equation.
3
Substitute the expression for xx into the first equation and solve for yy.
3(5−2y)−2y=7⇒15−6y−2y=7⇒15−8y=7⇒−8y=−8⇒y=13(5 - 2y) - 2y = 7 \Rightarrow 15 - 6y - 2y = 7 \Rightarrow 15 - 8y = 7 \Rightarrow -8y = -8 \Rightarrow y = 1
This step eliminates xx entirely, creating a single-variable linear equation that can be solved to find the exact value of yy.
4
Substitute the value of yy back into the isolated xx equation to find xx.
x=5−2(1)⇒x=5−2⇒x=3x = 5 - 2(1) \Rightarrow x = 5 - 2 \Rightarrow x = 3
Now that we have the numerical value for yy, we can plug it back in to find the corresponding value for xx.
5
Evaluate the final expression x−yx - y using the calculated values.
3−1=23 - 1 = 2
The question asks specifically for the value of x−yx - y, not just the ordered pair.

Key Concept

To solve a system of linear equations, you can use substitution by isolating a variable in one equation and substituting it into the other, or use elimination by combining the equations to cancel out a variable.
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Topics

Solving Systems of Linear Equations

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