Problem 90 - Intermediate Algebra

Question

Solve each equation. $$ 3(x-2)+4=3 x-2 $$

Step-by-Step Solution

Answer

The equation is true for all real values of \(x\).

Step 1: Distribute the 3

Start by distributing the 3 to both terms inside the parentheses. This means you multiply 3 by \(x\) and 3 by \(-2\). The equation becomes:\[3x - 6 + 4 = 3x - 2\].

Step 2: Simplify the equation

Combine like terms on the left side of the equation. The like terms are \(-6\) and \(+4\), which add up to \(-2\). This gives us:\[3x - 2 = 3x - 2\].

Step 3: Examine both sides of the equation

Notice that both sides of the equation are identical: \(3x - 2 = 3x - 2\). This indicates the equation is an identity.

Step 4: Conclude with the type of solution

Since each side of the equation is the same, the equation is an identity, meaning it is true for all values of \(x\). Thus, the solution set is all real numbers.

Key Concepts

Algebraic Identities

An essential concept in solving linear equations is understanding algebraic identities. An equation is known as an identity when each side of the equation is identical after simplification, regardless of the variable's value. In the provided example, the equation simplifies to \(3x - 2 = 3x - 2\).

• Both sides have the same expression, which indicates that every possible value of \(x\) will satisfy the equation.
• This type of equation is called an identity.
• It implies that the solution set is all real numbers.

Identifying an equation as an identity can often save time when solving problems, as you do not need to find specific values for the variable. Instead, you can simply conclude that any real number will satisfy the equation.

Distributive Property

The distributive property is a fundamental property used in simplifying and solving equations. It helps distribute a single term across terms inside a set of parentheses. In our exercise, we applied the distributive property to the expression \(3(x-2)\).

• We do this by multiplying 3 by both \(x\) and \(-2\).
• This results in \(3x - 6\).

This property is particularly useful in breaking down complex expressions, making them easier to work with.
The distributive property can be expressed generally as \(a(b+c) = ab + ac\). Understanding and applying this property allows for more straightforward manipulation and simplification of algebraic expressions.

Equation Simplification

Simplifying equations is a key step in solving them efficiently. It involves combining like terms and eliminating unnecessary components to focus on the main variables and constants.
In the example given, after applying the distributive property, you combine the constants \(-6\) and \(4\) to further simplify the equation to \(3x - 2 = 3x - 2\).

• Combining like terms helps in reducing the equation to its simplest form.
• This allows for easier comparison of both sides of the equation.
• Through simplification, you can detect patterns like identities or inconsistencies.

Streamlining equations by removing redundancies and clarifying expressions ensures that solving them becomes a more straightforward and less error-prone task.