Problem 38 - Algebra A Combined Function

Question

Solve each equation. See Examples 9 and \(10 .\) \(6 x-4=-2 x-10\)

Step-by-Step Solution

Answer

The solution is \(x = -\frac{3}{4}\).

Step 1: Move the variables to one side

Begin by adding \(2x\) to both sides of the equation to move all the \(x\) terms to one side. This gives you:\[6x + 2x - 4 = -10\]Which simplifies to:\[8x - 4 = -10\]

Step 2: Move the constants to the other side

Next, add \(4\) to both sides to move the constant term to the other side of the equation:\[8x - 4 + 4 = -10 + 4\]This simplifies to:\[8x = -6\]

Step 3: Solve for x

Now, divide both sides by \(8\) to solve for \(x\):\[x = \frac{-6}{8}\]Simplify the fraction:\[x = -\frac{3}{4}\]

Key Concepts

Understanding Linear Equations

Linear equations are fundamental in algebra and are equations of the first degree, meaning that the highest power of the variable is one. They generally have the standard form of \(ax + b = c\). In this format, \(x\) is the variable we are solving for, while \(a\), \(b\), and \(c\) are constants. The primary goal when working with linear equations is to find the value of \(x\) that makes the equation true, which involves manipulating the equation to isolate \(x\) on one side of the equation.
Linear equations can appear in various contexts, such as real-world problems ranging from simple budgeting cases to more complex mathematical modeling scenarios. Understanding how to handle these equations is crucial for solving problems in algebra as well as applying these skills in practical situations.
When solving linear equations, there are consistent steps that can be used, which make the process simpler and ensure a correct solution.

Basics of Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to make them easier to solve. It’s a skill that is essential in solving equations and understanding more about the relationships between different variables. In the case of linear equations, manipulation helps us move terms around to isolate the variable of interest.

There are a few key operations involved in algebraic manipulation:

• **Addition/Subtraction** – Used to move terms from one side of the equation to the other.
• **Multiplication/Division** – Used to simplify the equation and solve for the variable.

For example, in solving the original exercise, we added and subtracted terms in the equation to isolate the term with the variable \(x\), and then divided to solve for \(x\). Remember that whatever action you take on one side of the equation must also be done on the other side to maintain balance.
Mastering algebraic manipulation allows you to solve equations systematically and increase your understanding of how different parts of an equation interact.

Step-by-Step Process for Solving Equations

Solving linear equations involves a series of deliberate steps that simplify the equation and zero in on the solution. Here's how to approach such problems using the original exercise as an example:

**Step 1: Move the Variables to One Side**
This step involves transferring all terms with the variable to one side. For the given equation \(6x - 4 = -2x - 10\), we added \(2x\) to both sides to combine like terms, resulting in \(8x - 4 = -10\). This simplifies the equation.

**Step 2: Move the Constants to the Other Side**
Next, it's essential to isolate the variable term by moving constants to the opposite side. This is achieved by adding or subtracting constants on both sides. From our example, adding \(4\) to both sides punctuated the constants' movement, bringing us \(8x = -6\).

**Step 3: Solve for the Variable**
The final step is dividing by the coefficient of \(x\). This gives us an expression for \(x\). Continuing with our case, dividing both sides by \(8\), we derive \(x = -\frac{3}{4}\). Simplifying fractions is often necessary to present the answer in its most straightforward form.

Following these systematic steps ensures that equations are solved consistently and correctly, aiding in understanding core algebra concepts.