Problem 47 - Prealgebra

Question

Simplify each side of the following equations first, then solve. $$5 x+4 x+3 x=4-8$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{-1}{3}\).

Step 1: Combine Like Terms

Start by simplifying the left side of the equation by combining like terms. You have three terms involving \(x\): \(5x\), \(4x\), and \(3x\). Combine these terms to get \((5+4+3)x = 12x\). So, the equation becomes \(12x = 4 - 8\).

Step 2: Simplify the Right Side

Simplify the right side of the equation. You have \(4 - 8\), which simplifies to \(-4\). Now the equation is \(12x = -4\).

Step 3: Solve for x

To solve for \(x\), divide both sides of the equation by \(12\). This gives \(x = \frac{-4}{12}\), which simplifies to \(x = \frac{-1}{3}\).

Key Concepts

Combining Like Terms

In algebra, combining like terms is a fundamental skill that helps simplify expressions. Like terms are terms that have the same variables raised to the same powers. For example, in the equation given, the terms \(5x\), \(4x\), and \(3x\) are alike because they all contain the variable \(x\). When you combine these terms, you simply add their coefficients together.

Here's a step-by-step on how you do this:

• First, identify the like terms. In the example, it's clear that all terms contain the variable \(x\), so they can be combined.
• Next, add the coefficients: \(5 + 4 + 3 = 12\).

By combining like terms, we turned \(5x + 4x + 3x\) into \(12x\). This step makes simplifying and solving equations much more straightforward and manageable.

Simplifying Equations

The process of simplifying equations involves making them easier to understand and work with. By simplifying an equation, you can more easily identify the steps needed to solve it. In the given exercise, simplifying starts after combining like terms.

In the case of \(12x = 4 - 8\), the right side can be simplified further because \(4 - 8\) results in \(-4\). Let’s break down the step of simplifying:

• Look at the right-hand side, \(4 - 8\). Perform this basic arithmetic operation to get \(-4\).
• Replace the original expression on the right side of the equation with its simplified form. Hence, the equation becomes \(12x = -4\).

By simplifying the equation on both sides, you reduce complexity and set yourself up for a smoother path towards finding the unknown variable.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. It usually entails isolating the variable on one side of the equation. In our example, once the equation is simplified to \(12x = -4\), the next goal is to solve for \(x\).

Here’s the general process:

• First, identify the variable you need to solve for. In this instance, it's \(x\).
• To isolate \(x\), you'll need to get rid of the multiplication by \(12\). Do this by dividing both sides of the equation by \(12\).
• This division yields: \(x = \frac{-4}{12}\).
• Simplify \(\frac{-4}{12}\) to get \(x = \frac{-1}{3}\).

This technique of dividing to isolate the variable is key in solving linear equations. Once you find the value of \(x\), you've successfully solved the equation.