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Linear Equations in One Variable Flashcards

6 cards from real Bluebook SAT Test practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.

Read the first 6 Linear Equations in One Variable flashcards as text

  1. For what value of k does the equation (k² − 9)x = k − 3 have infinitely many solutions?

    Answer: k = 3

    Factoring the left side gives (k − 3)(k + 3)x = k − 3. When k = 3, both sides equal 0 for every value of x, producing infinitely many solutions. When k = −3, the left side is 0 but the right side is −6, giving no solution. Any other value of k yields a nonzero coefficient on x and exactly one solution.

  2. What is the solution set of the equation (2x + 3)/4 − (x − 1)/6 = (x + 5)/3?

    Answer: No real solution

    Multiplying every term by the LCD of 12 gives 3(2x + 3) − 2(x − 1) = 4(x + 5). Expanding: 6x + 9 − 2x + 2 = 4x + 20, which simplifies to 4x + 11 = 4x + 20. Subtracting 4x from both sides yields 11 = 20, a false statement, so no value of x satisfies the equation.

  3. If ax + b = cx + d, where a ≠ c, which of the following expresses x in terms of a, b, c, and d?

    Answer: x = (d − b) / (a − c)

    Collecting x terms on the left: ax − cx = d − b, so x(a − c) = d − b. Dividing both sides by (a − c) gives x = (d − b)/(a − c). Choice B is the negative of the correct answer. Choice C incorrectly adds rather than subtracts. Choice D inverts the fraction.

  4. Company A charges a $40 flat fee plus $0.15 per mile. Company B charges a $25 flat fee plus $0.20 per mile. At exactly how many miles driven do both companies charge the same total amount?

    Answer: 300 miles

    Setting the two cost expressions equal: 40 + 0.15m = 25 + 0.20m. Subtracting 0.15m from both sides gives 40 = 25 + 0.05m, then 15 = 0.05m, so m = 300. At 300 miles, both companies charge exactly $85 (Company A: $40 + $45; Company B: $25 + $60).

  5. The equation 5(x − a) = 3x + a has the solution x = 4. What is the value of a?

    Answer: 4/3

    First simplify the general equation: 5x − 5a = 3x + a → 2x = 6a → x = 3a. Since x = 4, we get 4 = 3a, so a = 4/3. Verify: 5(4 − 4/3) = 5(8/3) = 40/3 and 3(4) + 4/3 = 12 + 4/3 = 40/3. ✓

  6. The equation mx + n = p(x + q) has exactly one solution. Which of the following must be true?

    Answer: m ≠ p

    Rewriting the equation: mx + n = px + pq → (m − p)x = pq − n. For exactly one solution, the coefficient of x must be nonzero: m − p ≠ 0, i.e., m ≠ p. If m = p and n ≠ pq, the equation has no solution. If m = p and n = pq, the equation has infinitely many solutions. The relationship between n and pq is irrelevant as long as m ≠ p.