Problem 37 - Algebra 1

Question

Solve the proportion. Check for extraneous solutions. $$\frac{2}{3 t}=\frac{t-1}{t}$$

Step-by-Step Solution

Answer

Solve for \(t\) using the quadratic formula can give two possible values, and by substituting these calculated values of 't' back to the original expression, we can check for any possible extraneous solutions.

Step 1: Cross-multiply the fractions

Cross-multiply the fractions to get rid of the fraction and make the equation easy to solve. Doing so gives us: \(2t = 3t^2 - 3(t)\).

Step 2: Rearrange and simplify the equation

We can simplify the equation derived in step 1, to form a quadratic equation by bringing all terms to one side. Then, we can simplify the equation to \(3t^2 - 2t - 3 = 0\).

Step 3: Solve the quadratic equation

Solving the quadratic equation gives us points where the parabola crosses the x-axis i.e., the solutions. To solve the quadratic equation, we can use the quadratic formula which gives \(t = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}\), where \(a = 3\), \(b = -2\) and \(c = -3\). Substituting these values, we can obtain the values of \(t\).

Step 4: Check for extraneous solutions

Substitute the values obtained in the original equation to check if they are valid. If when substituting a particular value of 't' leads to division by zero or any undefined expression, then that is an extraneous solution and is disregarded.

Key Concepts

Proportion

A proportion is a mathematical equation that states two ratios are equal. In our problem, the proportion given is expressed as \(\frac{2}{3t} = \frac{t-1}{t}\). Proportions are commonly used in mathematics to solve problems involving rates, scales, and measurements.
A ratio like \(\frac{a}{b}\) compares two numbers, which can be used to express quantities like speed, density, or frequency.
In our specific problem, the proportion needs to hold true, meaning the cross-products of the equation should be equal. This becomes the starting point for solving the equation at hand.

Cross Multiplication

Cross multiplication is a method used to solve proportions. Essentially, you remove the fraction by multiplying diagonally across the equal sign.
This method simplifies the equation, making it easier to solve.
In our example, cross multiplying the equation \(\frac{2}{3t} = \frac{t-1}{t}\) gives:

• Multiply 2 by \(t\), the denominator of the second fraction, and
• Multiply \((t-1)\) by \(3t\), the denominator of the first fraction.

This results in the equation \(2t = 3t^2 - 3(t)\), which is simpler to handle without the fractions.

Quadratic Formula

The quadratic formula is a universal tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). After rearranging the equation to \(3t^2 - 2t - 3 = 0\), we can apply the quadratic formula.
The formula is expressed as: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substitute \(a = 3\), \(b = -2\), and \(c = -3\) into the formula to solve for \(t\). This calculation gives the possible values of \(t\) that satisfy the quadratic equation created from the original proportion.
This solution step helps find the points where a parabola intersects the x-axis in a quadratic graph.

Extraneous Solutions

Extraneous solutions are results that come from solving an equation, but do not satisfy the original equation. In this context, it's important to verify solutions in the original proportion to ensure they are valid.
If plugging the value of \(t\) back into the original equation causes a division by zero or another undefined condition, it is considered extraneous.
This step is crucial in equations involving variables in denominators, as simplification can introduce solutions that don't actually fit the conditions set by the original proportion.

• Always check back the potential solutions.
• Discard any results that do not satisfy the original equation, ensuring meaningful and correct final answers.

This verification step focuses on maintaining mathematical integrity and avoiding false solutions.