Problem 78 - Algebra and Trigonometry

Question

\(71-78\) Find the solution of the equation correct to two decimals. $$ \frac{1.73 x}{2.12+x}=1.51 $$

Step-by-Step Solution

Answer

The solution is \(x = 14.55\).

Step 1: Multiply Both Sides by the Denominator

Start by eliminating the denominator on the left side of the equation. Multiply both sides by \(2.12 + x\) to obtain:\[1.73x = 1.51(2.12 + x)\]

Step 2: Distribute the Right Side

Expand the right side of the equation by distributing the \(1.51\) across the terms inside the parentheses:\[1.73x = 1.51 \times 2.12 + 1.51x\]

Step 3: Simplify the Right Side

Calculate the multiplication to simplify the term:\[1.73x = 3.2012 + 1.51x\]

Step 4: Collect Like Terms

Move the term \(1.51x\) to the left side of the equation to get all \(x\) terms on one side:\[1.73x - 1.51x = 3.2012\]Simplify:\[0.22x = 3.2012\]

Step 5: Solve for x

To isolate \(x\), divide both sides by \(0.22\):\[x = \frac{3.2012}{0.22}\]Calculate the right side to find \(x\):\[x = 14.55\]

Step 6: Round to Two Decimal Places

Since \(x = 14.55\) is already in two decimal places, no further rounding is needed.

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves using different techniques to rearrange and simplify equations. The ultimate goal is often to solve for an unknown variable. Here are some key steps:

• Eliminate fractions by multiplying both sides by a common denominator. This helps simplify the equation into a form that's easier to work with.
• Distribute multiplication over addition to remove parentheses. For instance, if you have a term like \(a(b + c)\), distribute \(a\) by multiplying it with both \(b\) and \(c\).
• Move terms to one side of the equation using addition or subtraction. This helps to isolate the variable you're solving for.
• Combine like terms to simplify further. Like terms are terms that have the same variables raised to the same power, such as \(5x\) and \(3x\).

By mastering these techniques, you open up exciting possibilities in problem solving. Always check your transformations to ensure they're valid.

Fractions in Equations

Working with fractions in equations can initially seem challenging, but it's all about balance. Here's how to handle them effectively:

• Clear the fraction by multiplying every term in the equation by the denominator. This eliminates the fraction and makes working with the equation simpler.
• Watch for combined fractions. If a fraction appears in a denominator of another fraction, handling them can involve what’s called cross-multiplication.
• Make sure the equation remains balanced. Whatever operation you perform on the left, do the same on the right to maintain equality.
• Consider converting fractions to decimals if that could simplify your calculations. Be careful with accuracy, though.

Dealing with fractions is more about understanding and maintaining equation balance. With a bit of practice, manipulating these can become quite intuitive.

Decimal Precision

Decimal precision plays a crucial role in many algebra problems, especially when certain mathematical operations can yield lengthy decimal numbers. Here are some tips:

• Round the final answer to the required number of decimal places only at the last step. This avoids rounding errors during calculations.
• Use a calculator for precise computation when involving decimals, as manual calculation can introduce errors easily.
• Understand rounding rules: if the digit following your rounding point is 5 or higher, round up. Otherwise, round down.
• In algebraic problems, maintaining consistent decimal places ensures your solution remains accurate and within any specified tolerance.

Rounding too early or inconsistently can lead to errors, so maintaining decimal precision through the problem-solving process is key.