Solve the equations and determine if they have one solution, no solution, or infinite solutions.

Question image

Understand the Problem

The question is asking to solve a series of multi-step equations and categorize each one based on whether it has one solution, no solution, or infinite solutions.

Answer

A: $-\frac{4}{5}$, B: $\frac{17}{3}$, C: $\frac{7}{4}$, D: No solution, E: $\frac{21}{4}$, F: $\frac{23}{4}$, G: Infinite solutions, H: $\frac{3}{4}$, I: $\frac{43}{32}$.
Answer for screen readers
  • A: $x = -\frac{4}{5}$
  • B: $x = \frac{17}{3}$
  • C: $x = \frac{7}{4}$
  • D: No solution
  • E: $x = \frac{21}{4}$
  • F: $x = \frac{23}{4}$
  • G: Infinite solutions
  • H: $x = \frac{3}{4}$
  • I: $x = \frac{43}{32}$

Steps to Solve

  1. Equation A: Solve $-7x - 4 = 8x + 8$

    Start by isolating the variable on one side. Move the $8x$ to the left side:

    $$ -7x - 8x = 8 + 4 $$

    This simplifies to:

    $$ -15x = 12 $$

    Divide both sides by -15:

    $$ x = -\frac{12}{15} = -\frac{4}{5} $$

    This has one solution: $x = -\frac{4}{5}$.

  2. Equation B: Solve $x + 1 = 18 - 2x$

    First, bring all $x$ terms to one side:

    $$ x + 2x = 18 - 1 $$

    This simplifies to:

    $$ 3x = 17 $$

    Divide by 3:

    $$ x = \frac{17}{3} $$

    This has one solution: $x = \frac{17}{3}$.

  3. Equation C: Solve $2(x - 2) = 3 - 2x$

    Distribute the left side:

    $$ 2x - 4 = 3 - 2x $$

    Add $2x$ to both sides:

    $$ 4x - 4 = 3 $$

    Then add 4:

    $$ 4x = 7 $$

    Divide by 4:

    $$ x = \frac{7}{4} $$

    This has one solution: $x = \frac{7}{4}$.

  4. Equation D: Solve $2x - 4 = 2x + 15$

    Subtract $2x$ from both sides:

    $$ -4 = 15 $$

    This is a contradiction, so it has no solution.

  5. Equation E: Solve $5(5x + 7) = 140 + 5x$

    Distribute:

    $$ 25x + 35 = 140 + 5x $$

    Subtract $5x$ from both sides:

    $$ 20x + 35 = 140 $$

    Now, subtract 35:

    $$ 20x = 105 $$

    Divide by 20:

    $$ x = \frac{105}{20} = \frac{21}{4} $$

    This has one solution: $x = \frac{21}{4}$.

  6. Equation F: Solve $4x - 10 = 40 - 2(2x + 2)$

    Start by simplifying the right side:

    $$ 4x - 10 = 40 - 4x - 4 $$

    Combine terms:

    $$ 4x - 10 = 36 - 4x $$

    Add $4x$ to both sides:

    $$ 8x - 10 = 36 $$

    Add 10:

    $$ 8x = 46 $$

    Divide by 8:

    $$ x = \frac{46}{8} = \frac{23}{4} $$

    This has one solution: $x = \frac{23}{4}$.

  7. Equation G: Solve $4(x - B) = 32$

    Divide both sides by 4:

    $$ x - B = 8 $$

    This simplifies to:

    $$ x = 8 + B $$

    This is variable dependent and has infinite solutions.

  8. Equation H: Solve $2(4x + 12) = 4(6x + 4) - 4$

    Distributing on both sides:

    $$ 8x + 24 = 24x + 16 - 4 $$

    Simplifying right side:

    $$ 8x + 24 = 24x + 12 $$

    Rearranging:

    $$ 8x - 24x = 12 - 24 $$

    This simplifies to:

    $$ -16x = -12 $$

    Hence,

    $$ x = \frac{12}{16} = \frac{3}{4} $$

    This has one solution: $x = \frac{3}{4}$.

  9. Equation I: Solve $4(8x - 15) = -32x + 26$

    Distribute on left side:

    $$ 32x - 60 = -32x + 26 $$

    Adding $32x$ to both sides:

    $$ 64x - 60 = 26 $$

    Add 60:

    $$ 64x = 86 $$

    Divide by 64:

    $$ x = \frac{86}{64} = \frac{43}{32} $$

    This has one solution: $x = \frac{43}{32}$.

  • A: $x = -\frac{4}{5}$
  • B: $x = \frac{17}{3}$
  • C: $x = \frac{7}{4}$
  • D: No solution
  • E: $x = \frac{21}{4}$
  • F: $x = \frac{23}{4}$
  • G: Infinite solutions
  • H: $x = \frac{3}{4}$
  • I: $x = \frac{43}{32}$

More Information

Each equation is categorized based on whether it yields one solution, no solution, or infinitely many solutions. Equations leading to contradictions or identities have been identified accordingly.

Tips

  • Misallocating terms when moving variables across the equal sign: Ensure sign changes are accurately applied.
  • Failing to simplify before solving may lead to complex equations that are harder to manage.
  • Misinterpreting infinite solutions: Check if the equation reduces to an identity (e.g., true for all values).

AI-generated content may contain errors. Please verify critical information

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