Problem 35 - Algebra and Trigonometry

Question

\(9- 46\) The given equation is either linear or equivalent to a linear equation. Solve the equation. $$ \frac{2 x-7}{2 x+4}=\frac{2}{3} $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{29}{2}\).

Step 1: Clear the Denominator

To clear the fractions, we'll use cross-multiplication. Multiply both sides by the denominators to get rid of the fractions: \[ 3(2x - 7) = 2(2x + 4) \]

Step 2: Distribute the Terms

Next, distribute the coefficients on both sides:\[ 6x - 21 = 4x + 8 \]

Step 3: Isolate the Variable

We want to keep the variable terms on one side and the constant terms on the other side. Subtract \(4x\) from both sides:\[ 6x - 4x - 21 = 8 \]This simplifies to:\[ 2x - 21 = 8 \]

Step 4: Solve for x

Now, solve for \(x\) by first adding 21 to both sides:\[ 2x = 29 \]Then, divide both sides by 2:\[ x = \frac{29}{2} \]

Key Concepts

Cross-Multiplication

Cross-multiplication is a powerful technique used to solve equations involving fractions. It eliminates the fractions by multiplying across the diagonal, thus simplifying the equation.
In the given exercise, we started with the equation:\[ \frac{2x-7}{2x+4} = \frac{2}{3} \]
To clear the fractions, we cross-multiply. This means we multiply the numerator of one fraction by the denominator of the other fraction. So, we have the equation transformed to:

• The left side: \(3(2x - 7)\)
• The right side: \(2(2x + 4)\)

This leads us to the equation:\[3(2x - 7) = 2(2x + 4)\]
This step ensures that the denominators are effectively "removed," providing a fresh starting point for solving the equation without fractions.

Variable Isolation

Variable isolation involves rearranging an equation to get the variable of interest by itself on one side. This process simplifies solving for that variable.
From the equation \(6x - 21 = 4x + 8\), our goal is to get all terms with \(x\) on one side. Here's how we do it:

• Subtract \(4x\) from both sides to remove \(x\) from the right side. This simplifies to \(6x - 4x = 8 + 21\).
• This further reduces to the equation \(2x - 21 = 8\).

By organizing terms, variable isolation helps to clearly see the arithmetic needed to find the value of \(x\). Our next step will further simplify the equation.

Equation Distribution

Distribution in mathematics involves multiplying each term inside a bracket with the term outside. It's a key step in simplifying equations.
For the equation \(3(2x - 7) = 2(2x + 4)\), distribution means:

• For the left side, multiply 3 by each term inside the bracket: \(6x - 21\).
• For the right side, multiply 2 by each term: \(4x + 8\).

After distribution, we have the new equation \(6x - 21 = 4x + 8\).
Distribution helps break down complicated expressions into simpler, workable pieces, making it easier to isolate and solve for variables.

Solving Fractions

Solving fractions often involves finding a common denominator or eliminating the fractions altogether, as we did using cross-multiplication.
In the provided exercise, the initial step of cross-multiplication set up an equation without fractions, simplifying the task to solve for \(x\).
This technique is valuable because it reduces errors and simplifies complex operations with cumbersome fractions. Solving equations with fractions often benefits from these strategies:

• Cross-multiplication, particularly useful in "cross" equations.
• Finding a least common denominator if the equation has multiple fractions with different denominators.
• Simplifying fractions wherever possible before proceeding with further algebraic steps.

Overall, managing fractions properly streamlines the process of solving equations, allowing for cleaner and more straightforward solutions.