Problem 21 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation using two methods. Then explain which method you prefer. $$\frac{2 x}{5}+5 x=\frac{4}{3}$$

Step-by-Step Solution

Answer

The two solutions for the equation \( \frac{2x}{5} + 5x = \frac{4}{3} \) are both approximately x = 0.2469 using two methods. It is a matter of preference on whether to use simplification and direct solution or cross multiplication, depending on the nature of the equation.

Step 1: Method 1: Simplify and Solve

First rewrite \( \frac{2x}{5} \) as \( 0.4x \), so we have \( 0.4x + 5x = \frac{4}{3} \). Simplifying, we find that \( 5.4x = \frac{4}{3} \). Dividing both sides by 5.4 to solve for x gives \( x = \frac{4}{3 * 5.4} = \frac{4}{16.2}=0.2469 \).

Step 2: Method 2: Cross Multiplication

Rewrite the equation in fraction form \( \frac{2x+25x}{5} = \frac{4}{3} \). Then cross multiply, 3(2x+25x)= 4*5. This simplifies to 81x = 20 or \( x = \frac{20}{81}= 0.2469 \).

Step 3: Method Comparison

Both methods gave the same result. However, Method 1 was perhaps simpler as it required fewer steps. On the other hand, Method 2 was more systematic and standard, making it less error-prone. Depending on the complexity of the equation, either method could be preferable.

Key Concepts

Cross Multiplication

Cross multiplication is a straightforward method useful for solving equations where variables appear in fractions. In our example, we first rewrite the original equation so that both sides are fractions:

• The left side becomes \( \frac{2x + 25x}{5} \) and the right side is \( \frac{4}{3} \).

From there, we can cross multiply, meaning we multiply the numerator of one side by the denominator of the other:

• Multiply 3 by \( (2x + 25x) \) and 4 by 5.
• This leads to the equation \( 3(27x) = 20 \).

Finally, solve the resulting equation to find \( x \). Cross multiplication is revered for being clear and structured, which minimizes errors.
This method is particularly helpful when dealing with proportion equations, as it allows us to handle fractions efficiently.

Simplification in Algebra

Simplification in algebra is all about transforming equations into simpler forms for easier understanding and solving. In our specific problem, simplification involves a few key steps:Start by rewriting each term or part of the equation in such a form that you can easily manipulate. Initially, \( \frac{2x}{5} \) was simplified to \( 0.4x \) to make computations friendlier.

• You then add like terms; combining \( 0.4x \) with \( 5x \) gives \( 5.4x \).
• The equation becomes much easier to solve after this step.

The core aspect of simplification is to make the equation as straightforward as possible without changing its meaning. This often involves combining like terms, factoring, or expanding algebraic expressions.
Simplification is fundamental since working with simpler forms helps avoid common errors and streamlines the problem-solving process.

Solving Linear Equations

Linear equations are those in which variables appear to the power of one. Solving them is a fundamental skill in algebra. In our exercise, after simplification, a single linear equation is set:

• \( 5.4x = \frac{4}{3} \)

To solve for \( x \), the coefficient of \( x \) must be isolated. This usually means performing operations like addition, subtraction, multiplication, or division on both sides of the equation:

• Divide both sides by 5.4 to isolate \( x \).
• This gives \( x = \frac{4}{16.2} \).

The beauty of solving linear equations lies in their straightforward nature and the universal methods applicable to them. Whether setting fractions equal through cross multiplication or simplifying step by step, linear equations follow a consistent pattern, aiding learners in developing a structured approach to problem-solving.