Problem 21 - Fundamentals of Mathematics

Question

Solve each equation. $$ 6 x+5=4 x-11 $$

Step-by-Step Solution

Answer

The solution is \( x = -8 \).

Step 1: Simplify the equation

To simplify the given equation, we need to get all terms with the variable on one side and constants on the other side. We start by subtracting \(4x\) from both sides of the equation: \[ 6x + 5 - 4x = 4x - 11 - 4x \] This simplifies to: \[ 2x + 5 = -11 \]

Step 2: Isolate the variable term

Next, we need to get \(x\) on its own by moving the constant term (5) to the other side. Subtract 5 from both sides of the equation:\[ 2x + 5 - 5 = -11 - 5 \] This simplifies to: \[ 2x = -16 \]

Step 3: Solve for the variable

Now, to solve for \(x\), we need to divide both sides by 2: \[ \frac{2x}{2} = \frac{-16}{2} \] This gives us: \[ x = -8 \]

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a crucial skill in solving linear equations like the one in this exercise. It involves rearranging and simplifying equations to make them easier to solve. This process includes operations such as addition, subtraction, multiplication, and division, which help us to both simplify and solve equations.

When we begin to solve an equation, our goal is to make it easier to handle. We do this by simplifying, which often means combining like terms. In the equation from our exercise, we simplified by subtracting terms like \(4x\) from both sides, making the equation more straightforward. This step is essential as it reduces the equation to a form where further simplification is possible.

• Combine like terms to simplify the equation.
• Perform same operations on both sides to maintain equality.

These techniques are fundamental to algebra and provide a solid foundation for solving more complex problems. Remember, each manipulation step should help bring the equation closer to a form where the solution becomes clear.

Variables

In algebra, variables are symbols that stand for unknown values. They allow us to write equations that describe a wide range of situations. In our exercise, the variable is \(x\), which we need to find the value of.

Variables are placeholders for numbers we don't yet know, and they are often represented by letters. The use of variables allows us to generalize mathematical relationships, solving real-world problems where the numbers may change but the relationships remain constant.

• Variables can represent any number within a given set.
• They are used to form expressions and equations.

Understanding how variables work is a key part of grasping algebraic concepts. It's important to learn how to manage them within equations, as they hold the key to solving the equation and finding the answer.

Isolating the Variable

Isolating the variable is one of the final steps in solving an equation. It involves manipulating the equation until the variable is by itself on one side of the equation, which allows us to determine its value.

In the example exercise, we aimed to get the variable \(x\) by itself. Initially, \(x\) was part of the term \(2x + 5\). We removed the constant term 5 by subtracting it from both sides of the equation, which helped us to isolate \(2x\). Eventually, we divided both sides by 2 to solve for \(x\).

• Move constant terms to one side and variable terms to the other.
• Divide or multiply to solve for the variable.

Always remember, isolating the variable is done by reversing operations. If something is added to \(x\), you subtract it, and if it’s multiplied, you divide. This principle is key in solving any algebraic equation, leading you towards the solution in a clear and systematic way.