If 3x + y = 19 and x + 3y = 1 for real numbers x and y, what is the value of the expression 2x + 2y?

Introduction / Context:
This question tests basic algebraic skills in solving simultaneous linear equations in two variables. Once x and y are found using the given equations, we substitute them into another expression, namely 2x + 2y. Such problems help students practise elimination or substitution methods and then evaluate a required combination of x and y rather than x and y individually.

Given Data / Assumptions:

• Two linear equations are given: 3x + y = 19 and x + 3y = 1.
• x and y are assumed to be real numbers.
• We need to compute 2x + 2y after solving the system.
• No additional constraints are imposed on x and y.

Concept / Approach:
The standard approaches for solving two linear equations in two variables are the substitution method and the elimination method. Here the elimination method is convenient. We manipulate the equations so that one variable cancels out when we add or subtract them. After finding x and y, we either directly compute 2x + 2y, or observe that 2x + 2y = 2(x + y) and first find x + y. Recognising such simplifications often saves time in exams.

Step-by-Step Solution:
Step 1: Write the equations clearly: (1) 3x + y = 19 and (2) x + 3y = 1.Step 2: Multiply equation (2) by 3 to align coefficients of x: 3(x + 3y) = 3 * 1 gives 3x + 9y = 3.Step 3: Subtract equation (1) from this new equation: (3x + 9y) - (3x + y) = 3 - 19.Step 4: This simplifies to 8y = -16, so y = -16 / 8 = -2.Step 5: Substitute y = -2 into equation (2): x + 3(-2) = 1, so x - 6 = 1, giving x = 7.Step 6: Now compute 2x + 2y = 2 * 7 + 2 * (-2) = 14 - 4 = 10.

Verification / Alternative check:
An alternative is to first compute x + y. From x = 7 and y = -2, we have x + y = 5. The expression 2x + 2y can be rewritten as 2(x + y). Thus 2(x + y) = 2 * 5 = 10, which matches the previous result. You may also substitute x = 7 and y = -2 back into the original equations to check that both are satisfied, confirming the correctness of the solution.

Why Other Options Are Wrong:
Option 20: This might come from incorrectly adding 3x + y and x + 3y without solving the system properly.Option 18: Could result from arithmetic mistakes when substituting values back into the expression.Option 11: May arise from adding 19 and 1 and then dividing by 2, which is not a valid method here.Option 14: Could come from taking only 2x with x = 7 and ignoring 2y, leading to an incomplete computation.

Common Pitfalls:
Students often make mistakes while multiplying or subtracting equations, leading to incorrect values of x or y. Another common error is to compute x and y correctly but then mishandle the final expression 2x + 2y, for example by forgetting the factor 2 or miscalculating with negative numbers. Carefully carrying out each arithmetic step and double checking substitutions can prevent such slips.

Final Answer:
The value of 2x + 2y is 10.