Question

Solve each of the given equations for \(x .\) Check your answers by hand or with a calculator $$5 x-4=3 x-6$$

Short answer

The solution is \(x = -1\).

Step-by-step solution

Identify the Equation

The given equation is \(5x - 4 = 3x - 6\). We need to solve for \(x\).

Isolate Terms with Variable

To isolate terms with \(x\), subtract \(3x\) from both sides of the equation. This gives us \(5x - 3x - 4 = 3x - 3x - 6\), which simplifies to \(2x - 4 = -6\).

Simplify by Removing Constant Terms

Add 4 to both sides to move the constant term. This gives \(2x - 4 + 4 = -6 + 4\), simplifying to \(2x = -2\).

Solve for the Variable

Divide both sides by 2 to solve for \(x\). This results in \(x = \frac{-2}{2}\), which simplifies to \(x = -1\).

Verify the Solution

Substitute \(x = -1\) back into the original equation to verify: \(5(-1) - 4 = 3(-1) - 6\). Simplifying both sides gives \(-5 - 4 = -3 - 6\), which results in \(-9 = -9\). This confirms that \(x = -1\) is correct.

Key concepts

Equation Solving

Equation solving is a fundamental skill in algebra. It involves finding the value of a variable that makes an equation true. In our exercise, “5x - 4 = 3x - 6”, we need to determine a value for \(x\) that balances both sides of the equation. We begin by examining the equation to see what is needed to make both sides equal. Understanding how to manipulate the equation by adding, subtracting, multiplying, or dividing terms is essential. This problem uses these operations to guide us to the solution efficiently. Remember, equations are like balances – whatever you do to one side, you must do to the other to keep it balanced.

Variable Isolation

Variable isolation refers to the process of manipulating an equation so that the variable ends up on one side by itself. In our equation, we want to isolate \(x\). Starting with \(5x - 4 = 3x - 6\), the initial step is to move all terms containing \(x\) to one side. We do this by subtracting \(3x\) from both sides. This adjustment leaves us with \(2x - 4 = -6\). Now, \(x\) terms are separated from constants. To further isolate \(x\), we need to clear it from any other terms - a crucial step in solving for the variable.

Verification Method

After solving for \(x\), it's important to verify our solution. Verification ensures that the solution not only fits our manipulated equation but also the original one. After isolating \(x\), we found \(x = -1\). We can check this by substituting \(-1\) back into the original equation, \(5(-1) - 4 = 3(-1) - 6\). Both sides of the equation simplify to \(-9\), confirming our solution is correct. Verification is a double-check to ensure no mistakes were made during manipulation.

Simplification Process

Simplification in algebra helps to reduce complexity in expressions or equations, making them easier to solve. In "5x - 4 = 3x - 6", once we gather like terms by moving \(3x\) and adding 4, simplifying results in \(2x = -2\). This less cluttered form is easier to work with, leading directly to the solution \(x = -1\). Simplification can occur at any step - it's an ongoing process ensuring expressions are as straightforward as possible for further manipulation or verification.