Problem 60 - Elementary and Intermediate Algebra

Question

Solve each proportion. $$ \frac{x-1}{9}=\frac{2 x}{3} $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{-1}{5}\).

Step 1: Cross-Multiply to Eliminate Fractions

In order to get rid of the fractions, we cross-multiply the terms of the equation. Multiply the numerator of the first ratio by the denominator of the second ratio, and vice versa. This gives us the equation: \[(x-1) imes 3 = 9 imes 2x\]

Step 2: Expand and Simplify

Expand the left-hand side of the equation:\[3(x-1) = 27 ext{x}\]When we distribute the 3 on the left, it becomes:\[3x - 3 = 18x\]

Step 3: Move Variables to One Side

Rearrange the equation to get all terms involving \(x\) on one side and the constant terms on the other. Subtract \(3x\) from both sides:\[-3 = 18x - 3x\] which simplifies to:\[-3 = 15x\]

Step 4: Solve for x

Divide both sides by 15 to solve for \(x\):\[x = \frac{-3}{15}\]Simplify the fraction:\[x = \frac{-1}{5}\]

Key Concepts

Understanding Cross-Multiplication

Cross-multiplication is a powerful technique for solving proportions. It's mainly used to eliminate fractions in an equation involving two ratios. Let's break it down with a common example you might see: solving the proportion \(\frac{a}{b} = \frac{c}{d}\). Instead of dealing with fractions, cross-multiplication allows us to convert the proportion into a simple equation without fractions.

This means you multiply the numerator of the first ratio by the denominator of the second ratio, and the numerator of the second ratio by the denominator of the first ratio. It turns our equation into \(a \times d = b \times c\).

• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Multiply the numerator of the second fraction by the denominator of the first fraction.
• Set the two products equal.

By cross-multiplying, you effectively "clear" the fractions, making it easier to solve the equation further. This technique helps to simplify and create a clear path towards finding the solution.

Steps to Solving Equations

Once you've eliminated fractions using cross-multiplication, you have to solve the resulting equation. The process involves finding the value of the variable that satisfies the equation. Let's look at the steps involved in solving equations:

• First, ensure that the equation is simplified as much as possible. This might involve expanding brackets or combining like terms.
• Next, you want to isolate the variable. This requires arranging the terms in the equation so all variables are on one side and constants on the other.
• Similarly, apply any necessary operations (such as addition, subtraction) to both sides of the equation to keep it balanced.

In the context of our original problem, after cross-multiplying, we ended up redesigning the equation to put all terms involving \(x\) on one side. By following these steps, you gradually narrow down the potential values of \(x\), getting closer to the answer.

The Role of Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging an equation to solve for a variable. It involves using algebraic operations such as addition, subtraction, multiplication, and division to simplify and solve equations.

In our equation, after cross-multiplying, we had \(3(x - 1) = 18x\). Through algebraic manipulation, we can:

• Expand the equation: Distribute the coefficients across terms within parentheses. This means multiplying \(3\) by each term inside the brackets, leading to \(3x - 3\).
• Combine like terms: Shift all terms involving the variable to one side and constant terms to the other.
• Simplify the equation: Reduce the equation to a simpler form by carrying out operations such as subtraction of \(3x\) from both sides, resulting in \(-3 = 15x\).

Once you have a simpler equation, solving for \(x\) becomes straightforward by dividing the remaining terms, leading to our final solution for the variable. This logical sequence simplifies seemingly complicated equations into much simpler expressions.