Linear equations with one, none or infinitely many solutions Knowledge-Based Practice

Questions

1. 1.

   Consider the linear equation $3(x + 4) - x = 2(x + 1) + c$, where $c$ is a real constant. Which of the following statements is true about this equation?

   • A) If $c = 10$, the equation has infinitely many solutions; if $c \neq 10$, it has no solution.
   • B) If $c = 10$, the equation has infinitely many solutions; if $c \neq 10$, it has exactly one solution.
   • C) If $c = 12$, the equation has infinitely many solutions; if $c \neq 12$, it has no solution.
   • D) If $c = 10$, the equation has no solution; if $c \neq 10$, it has infinitely many solutions.
2. 2.

   For real constants $a$, $b$, $c$, and $d$, which statement is always true about the linear equation $ax + b = cx + d$?

   • A) If $a = c$ and $b = d$, the equation has no solution.
   • B) If $a = c$ and $b \neq d$, the equation has no solution.
   • C) If $a \neq c$ and $b = d$, the equation has infinitely many solutions.
   • D) If $a \neq c$ and $b \neq d$, the equation has no solution.
3. 3.

   If the linear equation $a(x + 1) - 3 = 4x + b$ has infinitely many solutions, what is the value of $a - b$?

   • A) 1
   • B) 3
   • C) 5
   • D) 7
4. 4.

   The linear equation $4(x - 3) + 2 = 4x + k$ has no solution. Which of the following cannot be the value of the constant $k$?

   • A) $-14$
   • B) $-12$
   • C) $-10$
   • D) $0$
5. 5.

   Consider the linear equation $k(x - 5) = 2(x - 5)$, where $k$ is a constant. Which of the following statements is true?

   • A) If $k \neq 2$, the equation has no solution.
   • B) If $k = 2$, the equation has $x = 5$ as its only solution.
   • C) If $k = 2$, the equation has no solution.
   • D) If $k \neq 2$, the equation has exactly one solution, $x = 5$.
6. 6.

   If the linear equation $k^2 x - 4 = 9x + k - 1$ has no solution for $x$, what is the value of the constant $k$?

   • A) 3
   • B) -3
   • C) 9
   • D) -9
7. 7.

   Which expression can be added to the right-hand side of the equation $8x - 3 = 5x + 4$ so that the resulting equation has no solution?

   • A) $-3x$
   • B) -7
   • C) $3x$
   • D) $3x - 7$
8. 8.

   If the equation $2(ax - 1) = 8x - 2$ has infinitely many solutions and the equation $2(ax - 1) = bx + 5$ has no solution, what is the value of $a + b$?

   • A) 8
   • B) 12
   • C) 16
   • D) 10
9. 9.

   How many solutions does the linear equation $6(x - 2) + 12 = 4(x + 1) - 4$ have?

   • A) Infinitely many solutions
   • B) No solution
   • C) Exactly two solutions
   • D) Exactly one solution
10. 10.

    When solving a linear equation in one variable using valid algebraic operations, which outcome indicates that the equation has no solution?

    • A) The variable is isolated and equals zero, resulting in a statement of the form $x = 0$.
    • B) The variable terms cancel out completely, resulting in a false numerical statement such as $0 = 5$.
    • C) The variable terms cancel out completely, resulting in a true numerical statement such as $0 = 0$.
    • D) The variable terms on opposite sides have different coefficients, resulting in a statement such as $2x = 3x$.
11. 11.

    Which statement correctly describes the solution set of the linear equation $3(2x - 4) + 7 = 6x - 5$?

    • A) The equation has exactly one solution, which is $x = 0$.
    • B) The equation has exactly one solution, which is $x = 5$.
    • C) The equation has no solution because it simplifies to a contradiction.
    • D) The equation has infinitely many solutions because it simplifies to an identity that is true for all real numbers.
12. 12.

    For which condition on the constant $c$ does the equation $2(cx - 3) + 4 = 10x - 5$ have exactly one solution?

    • A) $c \neq 5$
    • B) $c = 5$
    • C) $c \neq -5$
    • D) $c = -5$
13. 13.

    For which values of the constants $m$ and $n$ does the equation $m(x - 2) + 6 = 4x + n$ have no solution?

    • A) $m = 4$ and $n = -2$
    • B) $m = 4$ and $n \neq -2$
    • C) $m \neq 4$ and $n = -2$
    • D) $m = -4$ and $n \neq 2$
14. 14.

    For what value of the constant $k$ does the equation $4(2x - 3) + 5 = kx - 7$ have infinitely many solutions?

    • A) $k = 8$
    • B) $k = 4$
    • C) $k = 2$
    • D) $k = -8$
15. 15.

    Which of the following equations has no solution?

    • A) $4(x + 2) = 2(2x + 4)$
    • B) $3x - 5 = 2x + 1$
    • C) $5(x - 1) + 2 = 5x + 3$
    • D) $2(3x - 1) = 6x - 2$

Answers and step-by-step solutions appear after you finish the test.