Problem 21 - Elementary Algebra

Question

Solve the equations. $$ 8 x+6=-10 $$

Step-by-Step Solution

Answer

Answer: The value of x in the given linear equation is -2.

Step 1: Subtract 6 from both sides

Subtract 6 from both sides of the equation to isolate the term containing the variable x: $$ 8x + 6 - 6 = -10 - 6 \\ 8x = -16 $$

Step 2: Divide by 8

Divide both sides of the equation by 8 to solve for x: $$ \frac{8x}{8} = \frac{-16}{8} \\ x = -2 $$ The solution for this equation is x = -2.

Key Concepts

Solving Equations

Solving equations is a fundamental concept in algebra that allows us to find the value of an unknown variable. This process involves determining the value that satisfies the equation, making it true when substituted back. In a basic linear equation like \(8x + 6 = -10\), our goal is to find the value of \(x\) that will balance both sides of the equation.
Each step in solving equations is designed to simplify the equation while maintaining its balance. Think of an equation like a seesaw; both sides must remain equal for it to "balance." By using techniques such as adding, subtracting, multiplying, or dividing both sides of the equation, we simplify until the variable is isolated. This process allows us to pinpoint the exact value that once inserted makes the equation correct.

Isolation of Variables

The isolation of variables is a crucial part of solving any equation. It means getting the variable we are trying to solve for, on one side of the equation by itself. Imagine that you need to unwrap a gift hidden under multiple layers of packaging to reveal what is inside - that's what we do in math when we isolate a variable.
In our original equation, \(8x + 6 = -10\), the variable \(x\) is initially surrounded by other numbers. Our task is to "unwrap" the 6 and the 8 from around \(x\).

• First, we remove the constant 6 from the left side by subtracting 6 from both sides of the equation, resulting in \(8x = -16\). This step moves us closer to revealing the value of \(x\).
• Then, we divide each side of the equation by 8, completely isolating \(x\), and find \(x = -2\).

By isolating variables effectively, we simplify the problem enough to see the solution clearly.

Algebraic Manipulation

Algebraic manipulation involves using various techniques to transform and simplify equations, making them easier to solve. This is a significant skillset in mathematics.
At the heart of manipulation are basic operations: addition, subtraction, multiplication, and division. Understanding how and when to use these operations enables us to methodically work through equations.
For instance, in the solution \(8x + 6 = -10\):

• We start by recognizing that subtraction can remove the constant term from one side of the equation, simplifying our work to \(8x = -16\).
• Next, division helps us handle the coefficient of \(x\). By dividing by 8, we strip away the multiplicative factor binding \(x\), revealing that \(x = -2\).

These careful, step-by-step manipulations ensure that the original equation stays balanced, while simultaneously uncovering the solution. Being able to perform algebraic manipulations confidently is like having a key to unlock solutions in mathematics.