Problem 2 - Elementary Algebra

Question

Solve each of the equations. $$\frac{x}{9}=\frac{5}{3}$$

Step-by-Step Solution

Answer

The solution is x = 15.

Step 1: Understand the Equation

The equation given is \( \frac{x}{9} = \frac{5}{3} \). This is a proportion where two fractions are set to be equal.

Step 2: Cross Multiply

To solve the proportion \( \frac{x}{9} = \frac{5}{3} \), we use cross-multiplication. Multiply both sides' numerators by the opposite side's denominator: \( x \times 3 = 5 \times 9 \).

Step 3: Simplify the Multiplication

Carry out the multiplication on both sides to simplify the equation: \( 3x = 45 \).

Step 4: Solve for x

To find \( x \), divide both sides of the equation \( 3x = 45 \) by 3: \( x = \frac{45}{3} \).

Step 5: Finalize the Solution

Simplify \( \frac{45}{3} \) to get \( x = 15 \).

Key Concepts

Proportions

A proportion is an equation that shows two fractions are equal. In its simplest form, a proportion can look like this: \( \frac{a}{b} = \frac{c}{d} \). What this tells us is that the ratio of \( a \) to \( b \) is the same as \( c \) to \( d \).

• Understanding proportions is essential for comparing quantities.
• In the equation \( \frac{x}{9} = \frac{5}{3} \), you see a real-life example of proportions.

To solve them, knowing the relationship between the parts of the fractions is key. Proportions are useful in various applications, like scaling recipes or converting measurements.
Start by acknowledging the proportion in an equation, which is often the first step in solving it. This concept forms the foundation to approach more complex problems.

Cross-Multiplication

Cross-multiplication is a method used to solve equations involving proportions. It is quite helpful when dealing with two fractions set equal to each other. The process involves multiplying the numerator of one fraction by the denominator of the other.

• In the equation \( \frac{x}{9} = \frac{5}{3} \), cross-multiplying results in \( x \times 3 = 5 \times 9 \).
• This simplifies the equation to get rid of the fractions entirely.

The term "cross" essentially refers to the "X" shape formed when you multiply across the fractions.
This technique transforms the equation into a simpler form, making it easier to solve for the unknown variable. It is reliable and quick for proportional equations.

Simplifying Fractions

Simplifying fractions is the process of reducing the fraction to its simplest form. After performing operations like multiplication or division, fractions often need to be simplified for clarity and accuracy. In our example, simplifying \( \frac{45}{3} \) gives you \( 15 \).

• Simplification involves dividing the numerator and the denominator by their greatest common divisor.
• This reduction results in the most understandable form of the fraction, which helps when interpreting the solution.

Effective simplification enhances problem-solving by providing clarity.
Students often find solutions become evident after this essential step. Remember that a simplified result is crucial in presenting your final answer.

Linear Equations

Linear equations are mathematical expressions that represent a straight line when graphed. They generally take the form \( ax + b = c \). In the context of proportions, transforming the equation \( \frac{x}{9} = \frac{5}{3} \) through cross-multiplication results in a linear equation: \( 3x = 45 \).

• To solve this, isolate the variable \( x \) by performing operations like addition, subtraction, multiplication, or division.
• Here, dividing both sides by 3 leads to \( x = 15 \).

Linear equations are fundamental in algebra due to their simplicity and wide array of applications.
Solving them often requires basic arithmetic and understanding the balance principle, which maintains equality throughout the process. Mastering these is pivotal for tackling more advanced algebraic problems.