Problem 32 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The solution to the proportion is \( u = \frac{3}{2} \). The value \( u = -1 \) is an extraneous solution and hence not valid.

Step 1 - Cross multiplication

To solve the proportion, the first step usually involves cross multiplication so as to equate the variables. Cross multiplying the proportion, \( \frac{u}{3} = \frac{1}{2u-1} \), gives: \( u(2u-1) = 3 \)

Step 2 - Simplifying

Now, simplify the left side of the equation, which should result in a quadratic equation: \( 2u^2 - u - 3 = 0 \)

Step 3 - Solving the quadratic equation

By factoring, solving this quadratic equation leads to: \( (2u - 3)(u + 1) = 0 \), which results in two potential solutions, \( u = \frac{3}{2} \) or \( u = -1 \)

Step 4 - Checking for extraneous solutions

Substitute these solutions into the original proportion to check if they are valid or extraneous. For \( u = \frac{3}{2} \), the original proportion becomes \( \frac{3/2}{3} = \frac{1}{2(3/2)-1} \), which simplifies to \( 1 = 1 \), confirming that this is a valid solution. For \( u = -1 \), the proportion becomes \( \frac{-1}{3} = \frac{1}{2(-1)-1} \), which results in \( -\frac{1}{3} = -1 \), indicating that this is an extraneous solution, and hence not a valid solution of the proportion.

Key Concepts

Cross Multiplication

Cross multiplication is a fundamental technique used to solve proportions, which are equations that show two ratios are equivalent. It involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the results equal to each other. The general form of a proportion is \( \frac{a}{b} = \frac{c}{d} \), which can be solved by cross multiplying to get \( ad = bc \).

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in a single variable \( x \), with the general form \( ax^2 + bx + c = 0 \) where \( a \), \( b \) and \( c \) are constants, and \( a \eq 0 \). Quadratic equations can be solved through various methods, including factoring, using the quadratic formula, or completing the square. The solutions are called roots and there can be two real roots, one real root (in the case of a repeated root), or two complex roots.

Extraneous Solutions

When solving equations, we sometimes find solutions that don't hold true when substituted back into the original equation; these are called extraneous solutions. They can occur for several reasons, such as squaring both sides of an equation, which can introduce false positives. In the context of proportions, one must always check the potential solutions to ensure they do not lead to division by zero or cause incorrect equality.

Factoring Quadratic

Factoring is a method to solve quadratic equations that involves expressing the quadratic polynomial as the product of two binomial expressions. This technique relies on finding a pair of numbers that not only add up to the coefficient of the linear term \( b \) but also multiply to the constant term \( c \) in the equation \( ax^2 + bx + c = 0 \). When successfully factored, the equation is set to zero, which allows us to apply the zero-product property to find the roots. Factoring is one of the simpler methods compared to others like the quadratic formula, but it only works when the quadratic can be neatly decomposed into factors.