Problem 70 - Algebra A Combined Function

Question

\(12 x+30+8 x-6=10\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{-7}{10}\).

Step 1: Combine Like Terms

Identify the terms that can be added together. In this equation, the terms with \(x\) are \(12x\) and \(8x\), and the constants are \(30\) and \(-6\). Add them together separately: \(12x + 8x = 20x\) and \(30 - 6 = 24\). This simplifies the equation to \(20x + 24 = 10\).

Step 2: Isolate the Variable Term

Subtract \(24\) from both sides of the equation to move the constant to the right side. This gives \(20x + 24 - 24 = 10 - 24\), simplifying to \(20x = -14\).

Step 3: Solve for the Variable

Divide both sides of the equation by \(20\) to solve for \(x\). So, \(x = \frac{-14}{20}\). Simplify the fraction to \(x = \frac{-7}{10}\).

Key Concepts

Simplifying Equations

When simplifying algebraic equations, the goal is to make the equation as straightforward as possible. To do this, we often look for opportunities to combine or "simplify" terms. Think of this process as tidying up an equation, making it easier to solve.
Here are a few easy steps to remember:

• Identify and group similar terms together. This means finding numbers that have the same variable or are both constants.
• Add or subtract these like terms to construct a simpler equation.

For instance, in the equation \(12x + 30 + 8x - 6 = 10\), we notice two groups: terms with \(x\) and constants. By simplifying these, we can combine them to form a clearer equation. After simplifying, your equation turns from a complex combination to something much more manageable!
Simplifying makes it easier in the following steps and creates a foundation for solving the equation.

Like Terms

Like terms are simply terms that have the same variables raised to the same power. Much like organizing items into similar groups, this step is crucial to simplifying and solving algebra equations.
Here’s how to spot and handle them:

• Identify terms with identical variables. In our example \(12x + 8x\), both are like terms because they contain the variable \(x\).
• Group and combine them by adding or subtracting their coefficients (the number in front of the variable). Here, you add \(12 + 8\) to get \(20x\).
• Similarly, constants like \(30\) and \(-6\) are like terms and can be combined through basic addition or subtraction. This combination gives us \(24\).

Using like terms allows for efficient simplification, giving rise to an equation stripped of its unnecessary parts and easier to handle.

Solving for Variables

Once the equation is simplified and like terms are combined, the next step is solving for the variable. This is where we find the value that makes the equation true.
Solving for variables effectively involves:

• Isolating the variable term on one side of the equation. This means getting all numbers and constants on one side and leaving the variable by itself on the other.
  • For example, \(20x + 24 = 10\) becomes \(20x = -14\) after subtracting \(24\) from both sides.
• Solving the equation by performing operations that make the variable stand alone. For instance, dividing \(20x = -14\) by \(20\) results in \(x = \frac{-14}{20}\).
• Finally, simplify the solution if possible to arrive at the most reduced form of the variable. Thus \(x = \frac{-7}{10}\).

When you solve for variables, you unravel the key to understanding the relationship within the equation and unlock the solution to the problem at hand.