Linear Equations in One Variable Flashcards

Solve for x: (2x + 1)/3 − (x − 2)/5 = (3x + 4)/15

Answer: x = −7/4

Multiply every term by the LCD = 15: 5(2x + 1) − 3(x − 2) = 3x + 4. Expand: 10x + 5 − 3x + 6 = 3x + 4, so 7x + 11 = 3x + 4. Subtract 3x and 11 from both sides: 4x = −7, giving x = −7/4. A common error is mishandling the subtraction of the second fraction's numerator, especially the sign on the −2 term inside the parentheses.

For what value of k does the equation 5 − 2(3x − k) = −6x + (k + 7) have infinitely many solutions?

Answer: k = 2

Expand the left side: 5 − 6x + 2k = −6x + k + 7. The −6x terms cancel from both sides, leaving 5 + 2k = k + 7, so k = 2. When k = 2, both sides simplify to −6x + 9, making the equation an identity true for every value of x — which is the condition for infinitely many solutions.

For what value of a does the equation (a + 3)x − 4 = 3(x + 1) + a have no solution?

Answer: a = 0

Expand the right side: (a + 3)x − 4 = 3x + 3 + a. Subtract 3x from both sides: ax − 4 = a + 3, so ax = a + 7. When a = 0, this becomes 0 = 7, a contradiction with no solution. Critical trap: when a = −7, the equation becomes 0 · x = 0, or 0 = 0 — that yields infinitely many solutions, not no solution. Students who confuse these two conditions will incorrectly choose a = −7.

Jake has $500 in a savings account and withdraws the same amount each week. After 6 weeks he has $290. Continuing at this rate, after how many total weeks will he FIRST have less than $100?

Answer: 12 weeks

Weekly withdrawal = (500 − 290) ÷ 6 = 35. Set up the strict inequality: 500 − 35w 400, so w > 11.43. Since w must be a whole number, the first week his balance drops below $100 is w = 12. Check: at w = 11, balance = 500 − 385 = $115 (still ≥ $100); at w = 12, balance = 500 − 420 = $80 (< $100). Choosing 11 is the classic off-by-one error from using ≤ instead of <.

Mia is currently 3 times as old as her sister. In 8 years, Mia will be exactly twice as old as her sister. How old is Mia right now?

Answer: 24 years old

Let the sister's current age = x, so Mia's current age = 3x. In 8 years: 3x + 8 = 2(x + 8) = 2x + 16. Solving: x = 8, so Mia is currently 3(8) = 24. Common traps: choosing 8 (the sister's age), choosing 32 (Mia's age in 8 years), or choosing 16 by forgetting to add 8 to the sister's age on the right side of the equation.

If 1/(x − 3) = 5/(2x − 1), what is the value of x?

Answer: 14/3

Cross-multiply: 1 · (2x − 1) = 5 · (x − 3), giving 2x − 1 = 5x − 15. Solving: −3x = −14, so x = 14/3. Always verify the solution doesn't cause a zero denominator: x − 3 = 14/3 − 9/3 = 5/3 ≠ 0, and 2x − 1 = 28/3 − 3/3 = 25/3 ≠ 0. A frequent error is making a sign mistake during distribution (5 · (−3) = −15, not +15), which produces x = 16/3.