Common Mistakes
Overview
This lesson identifies common errors students make when solving systems of linear equations and provides practice in recognizing and avoiding these mistakes.
Practice Problems
Question 1: A student solved y = 2x + 3 and y = x - 1 by setting 2x + 3 = x + 1. What mistake did they make?
Show Answer
Answer: They wrote x + 1 instead of x - 1
When substituting, you must copy the expression exactly. The correct setup is 2x + 3 = x - 1, giving x = -4.
Question 2: A student found x = 3 for the system 2x + y = 10 and x - y = 2. They then used x - y = 2 to get y = 3 - 2 = 1. Is this correct?
Show Answer
Answer: No, y = 1 is incorrect
From x - y = 2 with x = 3: 3 - y = 2, so y = 1 is actually correct. But let's check: 2(3) + 1 = 7, not 10. So x = 3 was wrong to begin with.
Question 3: When graphing y = 2x + 1, a student plotted (0, 1) and (2, 3). What's wrong?
Show Answer
Answer: The point (2, 3) is incorrect
When x = 2: y = 2(2) + 1 = 5, not 3. The correct second point is (2, 5).
Question 4: For the system x + y = 5 and 2x + 2y = 10, a student said there's no solution. Is this correct?
Show Answer
Answer: No, there are infinitely many solutions
The second equation is just twice the first, so they're the same line. Every point on x + y = 5 is a solution.
Question 5: A student substituted y = 3x into 2x - y = 5 and got 2x - 3x = 5, so x = -5. Check this work.
Show Answer
Answer: The work is correct, x = -5
2x - 3x = -x = 5, so x = -5. Then y = 3(-5) = -15. Check: 2(-5) - (-15) = -10 + 15 = 5. Correct!
Question 6: A student solved 3x + 2y = 12 and y = x - 1 by substituting to get 3x + 2(x - 1) = 12, then 3x + 2x - 1 = 12. Find the error.
Show Answer
Answer: Should be 2x - 2, not 2x - 1
When distributing 2(x - 1), you get 2x - 2. The correct equation is 3x + 2x - 2 = 12, so 5x = 14, x = 2.8.
Question 7: Two lines with equations y = 3x + 2 and y = 3x - 4 were graphed. A student said they intersect at one point. Is this true?
Show Answer
Answer: No, they are parallel and never intersect
Both lines have slope 3, so they're parallel. Different y-intercepts (2 and -4) mean they never meet. No solution exists.
Question 8: A student found the solution (4, 2) for x + y = 6 and 2x - y = 10. Verify if this is correct.
Show Answer
Answer: Incorrect - doesn't satisfy 2x - y = 10
Check: 4 + 2 = 6 (works). But 2(4) - 2 = 6, not 10. The correct solution is found by adding equations: 3x = 16, x = 16/3.
Question 9: When solving x = y + 2 and 3x - 2y = 8, a student substituted to get 3(y + 2) - 2y = 8, then 3y + 2 - 2y = 8. Find the error.
Show Answer
Answer: Should be 3y + 6, not 3y + 2
3(y + 2) = 3y + 6. The correct equation is 3y + 6 - 2y = 8, so y = 2 and x = 4.
Question 10: A student got x = 0 and y = 0 as the solution to 2x + 3y = 6 and 4x + 6y = 12. Is this correct?
Show Answer
Answer: No, (0, 0) doesn't satisfy either equation
Check: 2(0) + 3(0) = 0, not 6. The system has infinitely many solutions since the equations are equivalent, but (0, 0) isn't one of them.