Problem 37 - Elementary and Intermediate Algebra

Question

Solve each proportion. $$ \frac{x+1}{5}=\frac{3}{15} $$

Step-by-Step Solution

Answer

The solution to the proportion is \( x = 0 \).

Step 1: Understand the Proportion

A proportion means that two fractions are equal. Here, the equation is \( \frac{x+1}{5}=\frac{3}{15} \). The goal is to find the value of \( x \) that makes this equation true.

Step 2: Cross-Multiply to Eliminate Fractions

Cross-multiplying involves multiplying diagonally across the equals sign. Multiply \((x+1)\) by 15 and 5 by 3:\[ 15(x+1) = 5 \cdot 3 \]

Step 3: Simplify Both Sides of the Equation

Distribute the 15 on the left side of the equation and multiply on the right side:\[ 15x + 15 = 15 \]

Step 4: Solve for x

Subtract 15 from both sides to isolate \( 15x \):\[ 15x = 15 - 15 = 0 \]Then divide both sides by 15 to solve for \( x \):\[ x = \frac{0}{15} = 0 \]

Key Concepts

Understanding Cross-Multiplication

Cross-multiplication is a method used primarily to solve proportions, which are equations that signify two ratios are equal. It simplifies the process of eliminating fractions to easily solve for the unknown variable. By cross-multiplying, you multiply the numerator of each fraction by the denominator of the opposite fraction. This action results in an equation without fractions. Here's how it works in practice. Imagine you have a proportion like \( \frac{x+1}{5} = \frac{3}{15} \). Cross-multiplication involves multiplying \((x + 1)\) by 15 and 5 by 3, leading to the equation:

• \( 15(x + 1) = 5 \times 3 \)

This step effectively removes the fractions, allowing you to focus on simpler arithmetic. Cross-multiplication is a potent tool because it turns a complex fraction problem into a straightforward algebraic equation.

The Art of Solving Linear Equations

Solving linear equations is a fundamental skill in algebra, involving finding the value of the unknown variable that makes the equation true. After cross-multiplying, you are often left with a linear equation.To solve it, follow these steps:1. Simplify both sides of the equation, usually through operations like distributing multiplication over addition:

• For example, in the equation \( 15(x + 1) = 15 \), distribute the 15 to get \( 15x + 15 = 15 \).

2. Isolate the variable:

• You achieve this by performing subtraction or addition. Subtract 15 from both sides to simplify to \( 15x = 0 \).

3. Solve for the variable:

• Divide by the coefficient of the variable to solve for it. Here, dividing both sides by 15 gives \( x = 0 \).

Mastering these steps allows for solving various linear equations, paving the way for more advanced algebraic problem-solving.

Algebra Fundamentals: The Building Blocks

Algebra is a vital branch of mathematics that deals with symbols and the rules for manipulating these symbols. It's the science of operations and relations.Here are a few core concepts in algebra:

• Variables and Constants: Variables like \(x\) represent unknown values, while constants are known values like numbers.
• Expressions and Equations: Algebraic expressions combine numbers and variables using arithmetic operations, while equations indicate that two expressions are equal.
• Operations: Algebra uses operations such as addition, subtraction, multiplication, and division to manipulate equations.

Understanding these fundamentals helps you solve problems methodically. For example, in proportions or ratios, recognizing how to manipulate variables through cross-multiplication and solve equations algebraically is critical. These basics are indispensable tools that repeatedly come into play as mathematical complexity increases.