Select for a and for b values such that the equation x/a = x - b has more than one...

Phase 1: Owning the Dataset

Understanding the Equation

We need to find values of a and b such that the equation \(\mathrm{x/a = x - b}\) has more than one solution for x.

Visual Representation

For this algebraic problem, let's use equation format to track our transformations:

Original: \(\mathrm{x/a = x - b}\)
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Goal: Find a and b that give MORE THAN ONE solution

Phase 2: Understanding the Question

Key Insight About Solutions

For a linear equation in one variable, we can have:

• No solution (contradiction like \(\mathrm{0 = 5}\))
• Exactly one solution (typical case)
• Infinitely many solutions (identity like \(\mathrm{x = x}\))

"More than one solution" means infinitely many solutions, which happens when our equation becomes an identity.

Working with the Equation

Let's manipulate \(\mathrm{x/a = x - b}\) algebraically.

Multiplying both sides by a (assuming \(\mathrm{a ≠ 0}\)):
\(\mathrm{x = a(x - b)}\)
\(\mathrm{x = ax - ab}\)
\(\mathrm{x - ax = -ab}\)
\(\mathrm{x(1 - a) = -ab}\)

Critical Analysis

If \(\mathrm{a ≠ 1}\), we can divide by \(\mathrm{(1 - a)}\):
\(\mathrm{x = -ab/(1 - a)}\)
This gives exactly ONE solution - not what we want.

If \(\mathrm{a = 1}\), then:
\(\mathrm{x(1 - 1) = -1 × b}\)
\(\mathrm{x × 0 = -b}\)
\(\mathrm{0 = -b}\)
This means b must equal 0.

Phase 3: Finding the Answer

Verification

When \(\mathrm{a = 1}\) and \(\mathrm{b = 0}\):
\(\mathrm{x/1 = x - 0}\)
\(\mathrm{x = x}\)

This is an identity! It's true for ALL values of x, giving us infinitely many solutions.

Checking Our Answer Choices

From [-2, -1, 0, 1, 2]:

• For a: We need 1 tick mark
• For b: We need 0 tick mark

Both values are available in our choices.

Phase 4: Solution

Final Answer:

• Select 1 for a
• Select 0 for b

This makes the equation \(\mathrm{x/1 = x - 0}\), which simplifies to \(\mathrm{x = x}\), an identity true for all x values, giving us infinitely many solutions.