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Difficulty: HardSolving Systems of Linear Equations

Consider the following system of linear equations:

4(x−2)+y=3x 4(x - 2) + y = 3x
2x−3(y+1)=−2 2x - 3(y + 1) = -2

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

Answer: 2

Answer

2
The correct answer is derived by first simplifying both equations into standard form (x + y = 8 and 2x - 3y = 1). By applying the elimination method, we find the system's solution is x = 5 and y = 3. Finally, evaluating the specific requested expression gives us x - y = 5 - 3 = 2.

Step-by-Step Solution

1
Simplify the first equation.
x + y = 8
Equations must be in standard form to easily apply elimination or substitution.
2
Simplify the second equation.
2x - 3y = 1
Expanding the terms and grouping variables on one side creates the standard form.
3
Multiply the simplified first equation by 3.
3x + 3y = 24
This aligns the y-coefficients (3y and -3y) for the elimination method.
4
Add the modified first equation to the simplified second equation.
5x = 25
Adding the equations eliminates the y variable, allowing us to solve for x.
5
Divide both sides by 5.
x = 5
Isolates the variable x.
6
Substitute x = 5 into the simplified first equation (x + y = 8).
y = 3
This finds the value of the remaining variable, y.
7
Evaluate the target expression (x - y).
5 - 3 = 2
The question specifically asks for the difference between x and y, not just their individual values.

Key Concept

Solving systems of linear equations involving pre-simplification
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Topics

Solving Systems of Linear Equations

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