Problem 37 - Elementary Algebra

Question

Solve. $$ -x-2+4 x=5+3 x-7 $$

Step-by-Step Solution

Answer

The equation is an identity, true for any \(x\).

Step 1: Simplify Both Sides

Start by simplifying both sides of the equation. The left side is \(-x - 2 + 4x\), which simplifies to \((4x - x) - 2 = 3x - 2\). The right side of the equation is \(5 + 3x - 7\), which simplifies to \(3x - 2\). So the equation becomes: \(3x - 2 = 3x - 2\).

Step 2: Analyze the Simplified Equation

Look at the simplified equation: \(3x - 2 = 3x - 2\).Since both sides of the equation are identical, it indicates that the equation is true for any value of \(x\). This is an identity.

Key Concepts

Equation Solving

Solving an equation means finding the value of the variable that makes the equation true. In algebra, the goal is to isolate the variable, usually represented by a letter like "x". By manipulating the equation using valid mathematical operations, we can determine the unknown value.
The key steps in solving an equation typically involve:

• Simplifying each side of the equation individually.
• Rearranging terms to get all terms involving the variable on one side and constants on the other.
• Performing operations to isolate the variable. These might include addition, subtraction, multiplication, or division.

In the exercise, simplifying both sides was the crucial step. The equation initially had terms combined in a way that wasn't straightforward. Only after simplification did the nature of the equation become clear.

Identity Equation

An identity equation is a unique type of equation where both sides are identical, meaning every value for the variable makes the equation true. When simplified, identity equations look something like: "Something = Something," often involving identical expressions on both sides.
A simplified form was reached in the exercise where both sides of the equation "3x - 2 = 3x - 2" were the same.

• Since they are equal, it signifies that the initial equation holds for any value of the variable.
• This type of equation doesn’t have a single solution but infinite solutions, as any real number can satisfy the equation.
• Recognizing an identity can be helpful, as no further calculations are needed to find a specific solution.

Understanding identity equations is crucial, especially in algebra, as they indicate equality rather than solve for one particular answer.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves reducing complex expressions into more manageable forms. This process can often clarify what the equation represents and aid in the solving process.
Simplification involves:

• Combining like terms, which are terms that have the same variable raised to the same power.
• Using arithmetic operations like addition and subtraction to reduce the number of terms.
• Removing parentheses by correctly applying the distributive property.

In the given exercise, simplifying the expression by combining like terms was essential:

• The left side "-x - 2 + 4x" simplified to "3x - 2" by combining like terms, "4x" and "-x".
• Similarly, the right side "5 + 3x - 7" simplified to "3x - 2".

The simplification on both sides highlighted the equation's nature as an identity, demonstrating the importance of simplification in understanding and solving equations.