Problem 64 - Algebra 2

Question

PREREQUISITE SKILL Use cross products to solve each proportion. $$ \frac{4}{n}=\frac{6}{2 n-3} $$

Step-by-Step Solution

Answer

The solution is \( n = 6 \).

Step 1: Set Up Cross Multiplication

In solving proportions using cross-products, we multiply the numerator of each fraction by the denominator of the other fraction. For the given proportion \( \frac{4}{n} = \frac{6}{2n-3} \), we set up the cross products as follows: \[ 4(2n-3) = 6n. \]

Step 2: Expand the Equation

Now, let's distribute the 4 across the terms in the parentheses:\[ 4 \times 2n - 4 \times 3 = 6n, \]which simplifies to:\[ 8n - 12 = 6n. \]

Step 3: Solve for n

Move all terms involving n to one side of the equation and constants to the other side:\[ 8n - 6n = 12, \]which simplifies to:\[ 2n = 12. \]

Step 4: Divide to Find n

Finally, divide both sides of the equation by 2 to solve for n:\[ n = \frac{12}{2} = 6. \]

Key Concepts

Cross Multiplication

Cross multiplication is a fundamental technique used to solve proportions efficiently. A proportion is an equation stating that two ratios or fractions are equal. For example, if you have the proportion \( \frac{a}{b} = \frac{c}{d} \), cross multiplication involves multiplying the numerator of each fraction by the denominator of the opposite fraction. This gives the equation \( a \times d = b \times c \). By setting these products equal, cross multiplication allows you to transform the proportion into an algebraic equation, which can be more straightforward to solve for the unknown variable.

• Identify a proportion where two ratios are equal.
• Multiply the numerator of one fraction with the opposite fraction's denominator.
• Equate both products to form a simple equation.

Once you understand cross multiplication, solving proportions becomes a systematic process. This is particularly useful because it simplifies complex fractional equations into forms that are easier to analyze and solve.

Algebraic Equations

Algebraic equations are mathematical expressions involving unknown variables. They consist of terms that are often equated to show a relationship. In our context, after cross multiplying, we obtain an algebraic equation that needs solving. For example, from the cross multiplication step in the exercise, we get \( 4(2n-3) = 6n \). This is an algebraic equation where your goal is to isolate the variable on one side of the equation.

• The terms may include numbers, variables, and operations.
• The goal is to manipulate these terms to find the value of the unknown variable.
• They provide a systematic way to express mathematical relationships.

When you deal with algebraic equations, it’s essential to understand how to manipulate them by adding, subtracting, multiplying, or dividing both sides. This allows you to maintain equality while isolating the desired variable.

Solving for Variables

Solving for variables involves finding the value of an unknown quantity in an equation. Once you've set up an equation through cross multiplication, the next step is to find this unknown. In the worked-out example, after cross multiplying, we have \( 8n - 12 = 6n \). The procedure to solve equations often involves moving all terms containing the unknown to one side of the equation and all constant terms to the opposite side.

Steps:

• Combine like terms if necessary.
• Move all terms involving the variable to one side of the equation (e.g., \( 8n - 6n \)).
• Isolate the variable by performing appropriate arithmetic operations, such as division.

The aim is to isolate the variable on one side, leading to a straightforward expression, such as \( n = 6 \). Once you learn these techniques, solving equations by finding the value of unknowns becomes a clearer and more intuitive task.