Problem 77 - Algebra and Trigonometry

Question

\(71-78\) Find the solution of the equation correct to two decimals. $$ \frac{0.26 x-1.94}{3.03-2.44 x}=1.76 $$

Step-by-Step Solution

Answer

The solution is approximately 1.60.

Step 1: Eliminate the Fraction

Multiply both sides of the equation by the denominator \((3.03 - 2.44x)\) to eliminate the fraction. This gives:\[(0.26x - 1.94) = 1.76(3.03 - 2.44x)\].

Step 2: Expand the Right-Hand Side

Distribute \(1.76\) into \((3.03 - 2.44x)\):\[(0.26x - 1.94) = (1.76 \times 3.03) - (1.76 \times 2.44x)\].Calculate the products:\[(0.26x - 1.94) = 5.3328 - 4.2944x\].

Step 3: Collect x Terms on One Side

Add \(4.2944x\) to both sides to move all terms involving \(x\) to one side:\[0.26x + 4.2944x = 5.3328 + 1.94\].This simplifies to:\[4.5544x = 7.2728\].

Step 4: Solve for x

Divide both sides by \(4.5544\) to solve for \(x\):\[x = \frac{7.2728}{4.5544}\].Calculate the division to find:\[x \approx 1.597\].

Step 5: Round the Solution

Round the value of \(x\) to two decimal places:\[x \approx 1.60\].

Key Concepts

Understanding Linear Equations

A linear equation is simply an equation involving variables where the highest power of the variable is one. These equations are pretty straightforward because they represent straight lines when graphed. In our exercise, the equation is given in a different form due to the presence of a fractional term. But essentially, we treat it as a linear equation because the variable, in this case, is not raised to any power higher than one.

Linear equations are a fundamental part of algebra and are used extensively in various real-life applications. Solving them involves basic operations we use in everyday calculations like addition, subtraction, multiplication, and division.

• Always ensure the equation is balanced—whatever you do to one side, do to the other.
• Think of the equation as a balance scale; each side must have the same value to remain balanced.

This simplicity is why linear equations form the building block for more complicated algebraic concepts.

Breaking Down Fractional Expressions

Fractional expressions in equations can be a bit tricky because they involve fractions that might be fractions of variables or constants. In this exercise, we have a fractional equation where the entire expression on the left side is split by a denominating factor. To solve equations involving fractional expressions, a common technique is to eliminate the fraction.

Here’s how we do it:

• Identify the denominator and remove it by multiplying both sides of the equation.
• This action effectively 'clears 'out the fraction, transforming a potentially complex problem into a simpler equation.

An important note is that while multiplying by the denominator simplifies the equation, it also requires careful attention to avoid introducing errors. A single misstep can lead to incorrect solutions. Thus, maintain vigilance when employing this technique.

Rounding Numbers in Equations

Rounding is a process used to simplify numbers, making them easier to work with. Especially in mathematical solutions where an exact decimal may not be required or is too lengthy. The step-by-step solution of our original exercise concluded with rounding to two decimal places.

Here are some key points to remember about rounding:

• If the digit following your rounding point is 5 or greater, round up the last retained digit by one.
• If it’s less than 5, leave the last retained digit as it is.

This is an essential skill, particularly in handling solutions that require a specific degree of precision, like in financial calculations or measurements. This ensures consistency and correctness in presentation, especially when decimals are involved.

Step-by-Step Solving of Equations

When faced with an equation, taking a step-by-step approach is crucial to finding the correct solution. Here's a breakdown of the essential steps:

Firstly, clearly identify all parts of the equation. Then, eliminate any fractions by finding a common factor or by multiplying both sides by the denominator, as we tackled earlier. Once you have a neat equation free of fractions, focus on simplifying it by:

• Expanding expressions to clear any grouping symbols like parentheses.
• Collecting like terms to bring all variable terms to one side and constant terms to the other.
• Solve for the variable by isolating it—this might involve dividing both sides by a factor.

If an exact calculation is necessary, make sure to perform all arithmetic operations meticulously. As a final step, don't forget to apply rounding if the problem demands it. This structured method helps ensure accuracy and aids in understanding the mathematics involved.