Problem 42 - Prealgebra

Question

Simplify each side of the following equations first, then solve. $$4 x+5 x-8=6+4$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 2\).

Step 1: Simplify Left Side

Combine like terms on the left side of the equation: \(4x + 5x - 8\). This simplifies to \(9x - 8\).

Step 2: Simplify Right Side

Combine constants on the right side of the equation: \(6 + 4\). This simplifies to \(10\).

Step 3: Write Simplified Equation

With both sides simplified, the equation becomes: \(9x - 8 = 10\).

Step 4: Isolate Variable

Add 8 to both sides to isolate terms with \(x\): \(9x - 8 + 8 = 10 + 8\), which simplifies to \(9x = 18\).

Step 5: Solve for x

Divide both sides by 9 to solve for \(x\): \(x = \frac{18}{9}\). Simplify to find \(x = 2\).

Key Concepts

Combining Like Terms

When solving linear equations, one essential step is to combine like terms. Like terms are terms in the equation that have the same variable raised to the same power. For example, in the equation \(4x + 5x - 8\), both \(4x\) and \(5x\) are like terms because they are both attached to the variable \(x\). By combining these, you simplify the equation:

• Combine the coefficients of like terms: \(4x + 5x = 9x\).
• The resulting equation becomes: \(9x - 8\).

Breaking down the process helps in reducing complexity and prepares the equation for further simplification. Remember, only coefficients of like terms are added or subtracted, ensuring that variables remain correctly associated.

Isolating Variable

Isolating the variable is a crucial step in solving equations, as it involves getting the variable on one side of the equation by itself. Once you have simplified both sides of your equation, you want to focus on the side of the equation with the variable. In our example, the equation is simplified to \(9x - 8 = 10\).

• Add 8 to both sides to cancel out the \(-8\): \(9x - 8 + 8 = 10 + 8\).
• This operation simplifies to: \(9x = 18\).

Ultimately, the goal is to have the variable term on one side, making it possible to perform operations that will solve for the variable. Ensuring the variable is isolated is vital before division or multiplication that solves for the exact value. It makes solving straightforward and prevents mistakes.

Simplifying Equations

Simplifying equations is the process of making an equation cleaner and easier to work with. This involves combining like terms, but it also includes ensuring that all constant terms are simplified as well. For the right side:

• Combine constants: \(6 + 4 = 10\).
• You are left with a simplified equation: \(9x - 8 = 10\).

By simplifying, you reduce the elements in the equation, making it easier to read and solve. Simplification helps you identify the necessary operations needed to isolate the variable effectively. This step sets the stage for the final operations that solve the equation, ensuring clarity and accuracy at every step.