Problem 24 - Algebra and Trigonometry

Question

\(9- 46\) The given equation is either linear or equivalent to a linear equation. Solve the equation. $$ \frac{2 x-1}{x+2}=\frac{4}{5} $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{13}{6} \).

Step 1: Cross-multiply to eliminate fractions

Multiply both sides of the equation by the denominators \( x+2 \) and 5 to get rid of fractions. This gives \( 5(2x-1) = 4(x+2) \).

Step 2: Distribute and simplify both sides

Distribute the numbers on both sides of the equation: \( 10x - 5 = 4x + 8 \).

Step 3: Isolate the x-terms

Subtract \( 4x \) from both sides to isolate the \( x \)-terms on one side: \( 10x - 4x - 5 = 8 \), simplifying to \( 6x - 5 = 8 \).

Step 4: Solve for x

Add 5 to both sides to isolate the \( x \)-term: \( 6x = 13 \). Then divide both sides by 6 to solve for \( x \): \( x = \frac{13}{6} \).

Key Concepts

Understanding Cross-Multiplication

Cross-multiplication is a valuable method for solving equations with fractions, particularly when they involve proportions. The goal of cross-multiplication is to eliminate the fractions by multiplying both sides of the equation by the denominators. For instance, consider the equation \[ \frac{2x-1}{x+2} = \frac{4}{5} \]
To cross-multiply, multiply the left-hand side by the denominator of the right-hand side and vice versa. Therefore, multiply \[ 5(2x-1) \]and \[ 4(x+2) \]to get a new equation without fractions:

• Left side: Multiply the numerator on the left by 5.
• Right side: Multiply the numerator on the right by the whole denominator of the left.

This method simplifies complex fraction equations into simple linear equations that are easier to solve, paving the way for the next steps.

Mastering Solving Equations

Solving equations is about finding the value of the variable that makes the equation true. Once you've cross-multiplied and eliminated fractions, you proceed with simplifying and solving the equation. Start by distributing the multiply across sums or differences on each side: \[ 5(2x-1) = 4(x+2) \] becomes\[ 10x - 5 = 4x + 8 \]
Organizing the terms correctly is essential. This requires applying basic arithmetic operations:

• Distribution: Multiply each term in the parenthesis by the coefficient outside.
• Simplification: Combine like terms and make sure to handle the equation properly without errors.

The goal is to simplify the equation until it becomes straightforward to isolate the variable. Once simplified, move to isolating terms which involves variable and number separation.

Techniques for Isolation of Variables

Isolation of variables involves rearranging the equation to get the variable on one side by itself. This is crucial in linear equations, allowing you to determine the exact solution. In this example after simplifying, you have:\[ 10x - 5 = 4x + 8 \]
To isolate the variable, follow these steps:

• Subtract or add: Eliminate terms from one side by performing the operations on both sides. Here, subtracting \[ 4x \] gives \[ 6x - 5 = 8 \].
• Move constants: Add or subtract to move numbers to the opposite side. Add 5 to both sides for \[ 6x = 13 \].
• Divide: Finally, divide by the coefficient in front of the variable, here dividing by 6 gives the final value \[ x = \frac{13}{6} \].

Understanding this step-by-step process is essential in solving linear equations. It ensures accuracy and builds confidence to tackle more complex algebraic expressions.