Question

What is the value of \(x\) in the following equation? \(3 x+2=\frac{4}{3} x\) (A) \(-\frac{6}{5}\) (B) \(-\frac{6}{5}\) (C) \(\frac{4}{6}\) (D) \(\frac{7}{9}\)

Short answer

The value of \(x\) is -\frac{6}{5\).

Step-by-step solution

Set the Equation Equal to Zero

Start by moving all terms involving \(x\) to one side of the equation to set the equation equal to zero. Subtract \(\frac{4}{3}x\) from both sides: \(3x + 2 - \frac{4}{3}x = 0\).

Simplify the Equation

Combine like terms on the left-hand side of the equation. Convert \(3x\) into a fraction with a common denominator, \(\frac{9}{3}x - \frac{4}{3}x = 0 - 2\), simplifying to \(\frac{5}{3}x + 2 = 0\).

Isolate the Variable Term

To isolate the term with \(x\), subtract 2 from both sides of the equation: \(\frac{5}{3}x = -2\).

Solve for x

Divide both sides of the equation by \(\frac{5}{3}\) to solve for \(x\): \(x = -2 \times \frac{3}{5}\). This simplifies to \(x = -\frac{6}{5}\).

Key concepts

Solving Equations

The process of solving equations is fundamental in algebra. It involves finding the value of a variable that makes the equation true. Let's start by looking at a problem we want to solve: 1. **Understand the Equation**: Given the equation \(3x + 2 = \frac{4}{3}x\), we need to determine the value of \(x\) which makes this equation valid.
2. **Setup the Equation**: The first step is to isolate all terms involving \(x\) on one side. Subtract \(\frac{4}{3}x\) from both sides to obtain \(3x + 2 - \frac{4}{3}x = 0\).
3. **Simplify and Solve**: As we proceed, we simplify the equation to make it easier to find the value of \(x\) by moving and simplifying terms.The goal is for the equation to represent a simple statement where \(x\) equals a particular value. This process shows how different operations help in reaching a solution.

Combining Like Terms

Combining like terms is a necessary step to simplify equations and make them easier to solve. In our example: - **Identify Like Terms**: Here, we have \(3x\) and \(-\frac{4}{3}x\) on the left side of the equation.
- **Common Denominator**: To combine these efficiently, convert \(3x\) into \(\frac{9}{3}x\) so both terms have a common denominator.
- **Subtract Terms**: Perform the subtraction \(\frac{9}{3}x - \frac{4}{3}x\) to simplify it to \(\frac{5}{3}x\).By combining these terms, we create a simpler expression. This step reduces the complexity and allows a clear path to isolate the variable.

Isolating Variables

Isolating the variable is the step where we get \(x\) on one side by itself. This process involves: 1. **Removing Constants**: From the simplified equation \(\frac{5}{3}x + 2 = 0\), subtracting 2 from both sides to yield \(\frac{5}{3}x = -2\).
2. **Divide to Isolate**: Next, divide both sides by \(\frac{5}{3}\). This operation clears everything except \(x\) on one side of the equation.
3. **Simplifying Further**: Multiplying \(-2\) by the reciprocal of \(\frac{5}{3}\) gives \(-2 \times \frac{3}{5} = -\frac{6}{5}\).This final action gives us the value of \(x\), completing the solution. It demonstrates isolating the variable effectively by using inverse operations to undo everything around \(x\).