Problem 26 - Algebra and Trigonometry

Question

\(9- 46\) The given equation is either linear or equivalent to a linear equation. Solve the equation. $$ \frac{1}{t-1}+\frac{t}{3 t-2}=\frac{1}{3} $$

Step-by-Step Solution

Answer

The solution is \(t = \frac{8}{11}\).

Step 1: Find a Common Denominator

The fractional equation needs to have a common denominator to eliminate the fractions. The denominators are \((t-1)\), \((3t-2)\), and the least common multiple is \((t-1)(3t-2)\). Multiply every term by \((t-1)(3t-2)\) to eliminate the fractions.

Step 2: Eliminate Fractions

Multiply each term by \((t-1)(3t-2)\):\[(t-1)(3t-2) \cdot \frac{1}{t-1} + (t-1)(3t-2) \cdot \frac{t}{3t-2} = (t-1)(3t-2) \cdot \frac{1}{3}\]Simplifying each term, we get:\[3t-2 + t(t-1) = \frac{(t-1)(3t-2)}{3}\]

Step 3: Simplify the Equation

Simplify the left side of the equation:\[3t - 2 + t^2 - t = \frac{(t-1)(3t-2)}{3}\]Combine like terms:\[t^2 + 2t - 2 = \frac{(t-1)(3t-2)}{3}\]

Step 4: Simplify the Right Side

Expand the expression on the right:\[\frac{(t-1)(3t-2)}{3} = \frac{3t^2 - 2t - 3t + 2}{3}\]Simplify:\[\frac{3t^2 - 5t + 2}{3}\]

Step 5: Equate and Solve

Equate both sides of the equation:\[ t^2 + 2t - 2 = \frac{3t^2 - 5t + 2}{3} \]Multiply through by 3 to clear the fraction:\[3(t^2 + 2t - 2) = 3t^2 - 5t + 2\]Which simplifies to:\[3t^2 + 6t - 6 = 3t^2 - 5t + 2\]

Step 6: Solve for t

Bring all terms to one side:\[3t^2 + 6t - 6 - 3t^2 + 5t - 2 = 0\]Simplify:\[11t - 8 = 0\]Solve for \(t\):\[t = \frac{8}{11}\]

Key Concepts

Common Denominator

When dealing with fractional equations, finding a common denominator is crucial to simplifying and ultimately solving the equation. A common denominator allows you to combine fractions into a single term, which makes manipulation much easier.
In the example equation \(\frac{1}{t-1} + \frac{t}{3t-2} = \frac{1}{3}\), we need to find the least common multiple (LCM) of the denominators \((t-1)\), \((3t-2)\), and 3. By identifying the product \((t-1)(3t-2)\), we effectively create a common footing for all the fractions. Therefore, each term must be multiplied by this LCM in order to eliminate the denominators.

• Step 1: Write out all the fractions in the equation.
• Step 2: Identify the least common multiple of all denominators.
• Step 3: Multiply each fraction by the common denominator to eliminate it.

This transforms the problem into a linear equation that is easier to solve, as fractions are no longer part of the equation.

Fractional Equations

A fractional equation contains one or more fractions with variables in the numerator or denominator. This can complicate solving, because it's tougher to isolate the variable.
Fractional equations often appear in the form \(\frac{a}{b} = \frac{c}{d}\), where \(a\), \(b\), \(c\), and \(d\) contain variables. To simplify, you often need to eliminate the fractions by multiplying all terms by a common denominator.
Some useful strategies include:

• Find the least common multiple of the denominators.
• Multiply every term by this LCM to clear out fractions.
• After clearing the fractions, solve the resulting linear equation for the variable.

By transforming a fractional equation into a form without fractions, you essentially revert it to a familiar algebraic equation, paving the way for simpler solutions.

Simplifying Equations

Simplifying equations is an important step in solving, especially after clearing fractions. Simplification involves combining like terms and reducing complex fractions. This makes the equation easier to understand and manage.
Once the denominators are eliminated, focus on simplifying both sides of the equation. Take for example the expression \(3t - 2 + t^2 - t = \frac{(t-1)(3t-2)}{3}\). Here:

• Combine like terms on the left to streamline the expression, resulting in \(t^2 + 2t - 2\).
• Perform algebraic expansion on the right to simplify the expression further, \(\frac{3t^2 - 5t + 2}{3}\).
• Equate both sides, and solve as you would any standard algebraic equation.

The trick is to reduce complexity step by step, thereby honing in on the variable. Through systematic simplification, you end up with equations that are clean and solveable.