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If 3x + 4y - 11 = 18 and 8x - 6y + 12 = 6, what is the value of the expression 5x - 3y - 9?

Difficulty: Medium

Correct Answer: -9

Explanation:


Introduction / Context:
This question tests solving simultaneous linear equations and then evaluating a new linear expression using the solved values. A clean approach is to first simplify both equations into standard form, solve for x and y, and substitute into 5x - 3y - 9.


Given Data / Assumptions:

  • 3x + 4y - 11 = 18
  • 8x - 6y + 12 = 6
  • Find 5x - 3y - 9


Concept / Approach:
Convert each equation into ax + by = c form. Then solve using substitution or elimination. Finally compute the required expression exactly.


Step-by-Step Solution:

Step 1: From 3x + 4y - 11 = 18, add 11 to both sides: 3x + 4y = 29. Step 2: From 8x - 6y + 12 = 6, subtract 12 from both sides: 8x - 6y = -6. Step 3: Divide Step 2 by 2 to simplify: 4x - 3y = -3. Step 4: Solve 4x - 3y = -3 for x: 4x = -3 + 3y, so x = (-3 + 3y)/4. Step 5: Substitute into 3x + 4y = 29: 3((-3 + 3y)/4) + 4y = 29. Step 6: Multiply by 4: 3(-3 + 3y) + 16y = 116. Step 7: Expand: -9 + 9y + 16y = 116 => -9 + 25y = 116 => 25y = 125 => y = 5. Step 8: Then x = (-3 + 15)/4 = 12/4 = 3. Step 9: Compute 5x - 3y - 9 = 5*3 - 3*5 - 9 = 15 - 15 - 9 = -9.


Verification / Alternative check:
Check in 3x + 4y = 29: 3*3 + 4*5 = 9 + 20 = 29 correct. Check 8x - 6y = -6: 24 - 30 = -6 correct.


Why Other Options Are Wrong:

-18, 18, -27, 9: these come from arithmetic slips (wrong substitution, sign error in moving constants, or wrong final evaluation).


Common Pitfalls:
Forgetting to move -11 and +12 correctly, or not simplifying 8x - 6y = -6 before substitution.


Final Answer:
-9

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