Problem 44 - Prealgebra

Question

Simplify each side of the following equations first, then solve. $$5 x-6+3 x=-6-8$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -1\).

Step 1: Simplify the Left Side of the Equation

Combine the like terms on the left-hand side of the equation. The expression is \(5x + 3x - 6\). Add \(5x\) and \(3x\) to get \(8x\), resulting in \(8x - 6\).

Step 2: Simplify the Right Side of the Equation

Combine the constants on the right-hand side of the equation. The expression \(-6 - 8\) simplifies directly to \(-14\).

Step 3: Set the Simplified Equations Equal to Each Other

The equation now reads \(8x - 6 = -14\). Both sides are simplified and ready for solving.

Step 4: Isolate the Variable Term

To isolate the term with \(x\), add 6 to both sides of the equation to eliminate the \(-6\) on the left-hand side. This gives you \(8x = -8\).

Step 5: Solve for x

Divide both sides of the equation by 8 to solve for \(x\). This results in \(x = -1\).

Key Concepts

Simplifying Equations

Simplifying equations is the first crucial step in equation solving. It involves reducing each side of the equation into a simpler form, making it easier to work with. In our given exercise, the left side of the equation is initially expressed as \(5x - 6 + 3x\). Before attempting to solve the equation, we need to simplify each side as much as possible. This means looking for opportunities to combine terms, like adding or subtracting similar terms. On the right side, the expression \(-6 - 8\) can also be simplified. By turning these more complex expressions into simpler ones, we reduce the chance for errors and make solving for the variable a much more straightforward task. Remember that simplification is all about making the equation look neater and easier to handle.

Combining Like Terms

Combining like terms is an essential part of simplifying equations and is all about making the equation neat and easy to understand. Like terms are terms that have the same variable raised to the same power. In prealgebra, you'll often see exercises asking you to combine terms such as \(5x\) and \(3x\). Since both terms have the variable \(x\), they can be combined.

• In the example, \(5x + 3x\) simplifies to \(8x\).
• The constant terms like \(-6\) and \(-8\) on the right side combine directly to \(-14\).

By combining like terms, we achieve a tidier equation that’s much easier to solve. Reducing an equation to its simplest form can prevent confusion later on in the problem-solving process.

Isolating the Variable

Once the equation has been simplified, the next step involves isolating the variable. This means getting the variable, usually \(x\), by itself on one side of the equation. To do this, we perform the same operation on both sides of the equation. In algebra, it’s important to maintain balance. Our exercise takes us to the equation \(8x - 6 = -14\). We want to remove the \(-6\) from the left side. By adding 6 to both sides, we find that \(8x = -8\). Now, \(x\) is almost isolated except for the number 8, which is multiplying it.

• To fully isolate \(x\), we divide both sides by 8.

This gives us the solution \(x = -1\). Isolating the variable ensures we find the correct value for \(x\), completing the equation-solving process.

Prealgebra

Prealgebra is the foundation upon which more advanced mathematics is built. It's the starting point for understanding algebra, and it covers the basic concepts that are crucial for solving equations. Simplifying equations, combining like terms, and isolating the variable are all essential skills learned in prealgebra. In our example, all these skills were put into practice to solve for \(x\). Each topic helps break down an equation into manageable steps:

• Understand how to simplify complex expressions.
• Learn to identify and combine like terms effortlessly.
• Gain confidence in isolating and solving for variables.

Prealgebra not only prepares students for subsequent math courses but also enhances logical thinking and problem-solving abilities, making it an invaluable part of any math curriculum. By mastering these basic skills, students can easily grasp more complicated concepts in algebra and beyond.