Problem 41 - Algebra and Trigonometry

Question

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: Cross-Multiply

To solve the equation \( \frac{2x-7}{2x+4} = \frac{2}{3} \), we start by cross-multiplying to eliminate the fractions. This gives the equation \( 3(2x - 7) = 2(2x + 4) \).

Step 2: Distribute on Both Sides

Next, distribute the numbers on each side of the equation. We get \( 6x - 21 = 4x + 8 \).

Step 3: Move Variable Terms to One Side

Now, move all terms involving \( x \) to one side of the equation. Subtract \( 4x \) from both sides: \( 6x - 4x - 21 = 8 \). This simplifies to \( 2x - 21 = 8 \).

Step 4: Isolate the Variable

Add 21 to both sides to isolate \( 2x \): \( 2x = 29 \).

Step 5: Solve for the Variable

Finally, divide both sides by 2 to solve for \( x \): \( x = \frac{29}{2} \).

Key Concepts

Cross-Multiplication

When you encounter an equation involving fractions, such as \( \frac{2x-7}{2x+4} = \frac{2}{3} \), cross-multiplication is a handy technique to get rid of the fractions. This method involves multiplying the numerator of one fraction by the denominator of the other and setting the two products equal to each other.

Here’s how it works with our example:

• Multiply \( 3 \) (the denominator of the right fraction) by \( 2x - 7 \) (the numerator of the left fraction).
• Multiply \( 2 \) (the denominator of the left fraction) by \( 2x + 4 \) (the numerator of the right fraction).

After performing these multiplications, you'll have the equation \( 3(2x - 7) = 2(2x + 4) \).

This equation no longer contains fractions, making it easier to work with in subsequent steps.

Distributive Property

Once you've cross-multiplied and set up your equation, use the distributive property to simplify each side. The distributive property states that \( a(b+c) \) is equal to \( ab + ac \).

In the example \( 3(2x - 7) = 2(2x + 4) \), apply the distributive property to open up the brackets:

• Distribute \( 3 \) over \( 2x - 7 \), resulting in \( 6x - 21 \).
• Distribute \( 2 \) over \( 2x + 4 \), resulting in \( 4x + 8 \).

Now, you have a simplified equation: \( 6x - 21 = 4x + 8 \).

With the equation simplified, it's clearer how you should manipulate the expressions to isolate the variable.

Isolating Variables

With a simpler equation \( 6x - 21 = 4x + 8 \) on hand, the goal is to isolate the variable, meaning you want \( x \) to stand alone on one side.

Here's how you can achieve that:

• Move all the terms involving \( x \) to one side by subtracting \( 4x \) from both sides: \( 6x - 4x - 21 = 8 \), simplifying to \( 2x - 21 = 8 \).
• Eliminate the constant term \(-21\) by adding \( 21 \) to both sides: \( 2x = 29 \).

Now, you’ve isolated \( 2x \). To solve for \( x \), divide both sides by \( 2 \): \( x = \frac{29}{2} \).

Remember, isolation of variables involves carefully executing inverse operations to peel back each layer until the variable is on its own.