If (u - 1)/(t + 3) = 4/(t - 1), where t neq -3 and t neq 1, what is...

1. INFER the solution strategy

• We have a rational equation with two variables where we need to solve for one variable in terms of the other
• Since we have a proportion with t appearing in both denominators, cross multiplication will eliminate the fractions

2. SIMPLIFY by cross multiplying

• Cross multiply: \((\mathrm{u} - 1)(\mathrm{t} - 1) = 4(\mathrm{t} + 3)\)

3. SIMPLIFY by expanding both sides

• Left side: \((\mathrm{u} - 1)(\mathrm{t} - 1) = (\mathrm{u} - 1)\mathrm{t} - (\mathrm{u} - 1) = \mathrm{ut} - \mathrm{u} - \mathrm{t} + 1\)
• Right side: \(4(\mathrm{t} + 3) = 4\mathrm{t} + 12\)
• Equation becomes: \(\mathrm{ut} - \mathrm{u} - \mathrm{t} + 1 = 4\mathrm{t} + 12\)

4. SIMPLIFY by rearranging terms

• Move all terms with t to the left side and constants to the right:
• \(\mathrm{ut} - \mathrm{t} - 4\mathrm{t} = \mathrm{u} - 1 + 12\)
• \(\mathrm{ut} - 5\mathrm{t} = \mathrm{u} + 11\)

5. SIMPLIFY by factoring out t

• Factor t from the left side: \(\mathrm{t}(\mathrm{u} - 5) = \mathrm{u} + 11\)
• Divide both sides by \((\mathrm{u} - 5)\): \(\mathrm{t} = \frac{\mathrm{u} + 11}{\mathrm{u} - 5}\)

Answer: A. \(\mathrm{t} = \frac{\mathrm{u} + 11}{\mathrm{u} - 5}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors during the expansion or term collection steps.

A typical mistake is incorrectly expanding \((\mathrm{u} - 1)(\mathrm{t} - 1)\), getting something like \(\mathrm{ut} - 1\) instead of \(\mathrm{ut} - \mathrm{u} - \mathrm{t} + 1\). This leads to collecting the wrong terms and getting an expression like \(\mathrm{t}(\mathrm{u} - 1) = \mathrm{u} + 13\), which would give \(\mathrm{t} = \frac{\mathrm{u} + 13}{\mathrm{u} - 1}\). This may lead them to select Choice C \(\left(\mathrm{t} = \frac{\mathrm{u} + 11}{\mathrm{u} - 1}\right)\) since it has the same denominator pattern as their incorrect work.

Second Most Common Error:

Poor INFER reasoning about the solution strategy: Some students try to solve this by isolating terms without cross multiplying first.

They might attempt to move terms around in the original fraction form, leading to confusion and computational dead-ends. This causes them to get stuck and randomly select an answer or resort to plugging in answer choices without systematic solution.

The Bottom Line:

This problem requires careful algebraic manipulation through multiple steps. Success depends on systematic cross multiplication followed by methodical expansion and term collection - any shortcuts or rushed algebra will derail the solution.