Problem 82 - Algebra and Trigonometry

Question

Find the solution of the equation rounded to two decimals. \(\frac{1.73 x}{2.12+x}=1.51\)

Step-by-Step Solution

Answer

The solution is approximately 14.55.

Step 1: Cross Multiply to Eliminate Fraction

Start by cross-multiplying to eliminate the fraction. Multiply both sides of the equation by \(2.12 + x\) to get rid of the denominator: \[1.73x = 1.51(2.12 + x)\].

Step 2: Expand the Right Side

Distribute \(1.51\) on the right side: \[1.73x = 1.51 \times 2.12 + 1.51x\] Which simplifies to \[1.73x = 3.2012 + 1.51x\].

Step 3: Isolate Variable Terms on One Side

Move the \(1.51x\) term from the right side to the left side by subtracting \(1.51x\) from both sides: \[1.73x - 1.51x = 3.2012\]. Simplify the left side: \[0.22x = 3.2012\].

Step 4: Solve for x

Divide both sides by \(0.22\) to solve for \(x\): \[x = \frac{3.2012}{0.22}\]. Calculate the division: \[x \approx 14.55\].

Step 5: Round Result to Two Decimals

Round the solution to two decimal places as requested, which is already done in the previous calculation.

Key Concepts

Cross-Multiplication

Cross-multiplication is a helpful technique when you want to solve equations involving fractions. It is particularly useful for clearing fractions from an equation to make it simpler and easier to handle. In essence, you multiply the numerator of one fraction by the denominator of the other fraction and vice versa. This way, the fractions are removed, leaving you with a basic algebraic equation to solve.
In the given exercise, we use cross-multiplication on the equation \( \frac{1.73x}{2.12 + x} = 1.51 \). To apply cross-multiplication, we do the following:

• Multiply each side of the equation by the cross product of the denominators, \( 2.12 + x \).
• This results in: \( 1.73x = 1.51 (2.12 + x) \).

Cross-multiplication effectively eliminates the fraction in the equation, enabling you to focus on solving a simpler, linear equation next.

Solving Equations

Solving algebraic equations means finding the value of the variable that makes the equation true. After cross-multiplying in our exercise, we have the equation: \( 1.73x = 1.51 \times 2.12 + 1.51x \). Our goal is to isolate the variable, \( x \), on one side of the equation.
We can tackle this through several steps:

• Distribute the constant: Distribute \(1.51\) to both terms inside the bracket: \(1.73x = 3.2012 + 1.51x\).
• Isolate the variable: Move \(1.51x\) from the right side to the left by subtracting it from both sides: \(1.73x - 1.51x = 3.2012\).
• Simplify: Calculate \(1.73x - 1.51x\) which yields \(0.22x\).
• Solve for \(x\): Divide both sides by \(0.22\) to determine \(x\): \(x = \frac{3.2012}{0.22}\).

Performing these operations will lead you to the solution for the unknown variable \(x\).

Rounding Decimals

Rounding decimals is a crucial step when a problem specifies a particular level of precision. In many cases, including economics, science, and engineering, results need to be rounded to a specified number of decimal places to ensure accuracy and ease of interpretation. In solving the equation \( x = \frac{3.2012}{0.22} \), we calculate and get \( x \approx 14.55 \) before rounding.
Here are helpful pointers on how to round decimals:

• Identify the digit at the required decimal place (the second decimal place for this exercise).
• Look at the digit immediately to the right of it.
• If that digit is 5 or greater, increase the identified digit by 1; if it is less than 5, leave it unchanged.

Applying this to our solution, \( 14.5454545 \ldots \) rounds up to \( 14.55 \) as stated in the solution. This ensures that your final answer matches the requested precision.