Problem 69 - Intermediate Algebra

Question

Solve each equation. If the equation is an identity or a contradiction, so indicate. See Example 9. $$ 2(x-3)=\frac{3}{2}(x-4)+\frac{x}{2} $$

Step-by-Step Solution

Answer

The equation is an identity, true for all values of \( x \).

Step 1: Distribute the coefficients

We start by distributing the coefficients into the parentheses on both sides of the equation. On the left side: \[ 2(x-3) = 2x - 6 \] On the right side: \[ \frac{3}{2}(x-4) + \frac{x}{2} = \frac{3}{2}x - 6 + \frac{x}{2} \]

Step 2: Simplify the right side

Combine the terms on the right side of the equation. The terms \( \frac{3}{2}x \) and \( \frac{x}{2} \) can be combined:\[ \frac{3}{2}x + \frac{x}{2} = \left(\frac{3}{2} + \frac{1}{2}\right) x = 2x \]Then the simplified right side becomes:\[ 2x - 6 \]

Step 3: Set both sides equal

Equating the simplified forms of both sides, we have:\[ 2x - 6 = 2x - 6 \]This equality holds true for all values of \( x \).

Step 4: Identify the nature of the equation

Since both sides of the equation are identical regardless of the value of \( x \), the equation represents an identity. An identity is true for all values within the domain of the equation.

Key Concepts

Identity Equations

An identity equation is a special type of equation where the expressions on both sides are identical for all possible values of the variables involved. In simpler terms, an identity holds true no matter what value you substitute for the variable. For example, if you have an equation like

• \(2x - 6 = 2x - 6\)

This equation is true regardless of the value of \(x\), because both sides will always evaluate the same, showing that it is an identity. Not all equations behave this way, but when you come across one, it indicates a fundamental equivalence between the expressions. This concept is incredibly important in algebra, as it shows us situations where both sides of an equation describe the same mathematical relationship.

Identifying identity equations is straightforward:

• After simplifying both sides, compare them. If they look identical, it's an identity!
• They are useful for verifying complex algebraic expressions and ensuring mathematical consistency.

Distributive Property

The distributive property is a very useful algebraic property that helps us to multiply expressions in a clear systematic way. This property allows us to remove parentheses by distributing a multiplier across the terms inside the parentheses. For example, in the expression:

• \(2(x-3)\)

Using the distributive property, you would multiply 2 by both \(x\) and -3:

• \(2 \cdot x + 2 \cdot (-3) = 2x - 6\)

This is a simplified expression without parentheses.

This property applies similarly on the right side of equations too, as in:

• \(\frac{3}{2}(x-4)\)
• \(\frac{3}{2} \cdot x + \frac{3}{2} \cdot (-4)\)

Which simplifies to:

• \(\frac{3}{2}x - 6\)

The distributive property is essential when solving linear equations because it transforms more complex expressions into simpler ones, making them easier to solve.

Simplifying Expressions

Simplifying expressions is the process of making an equation or expression easier to work with by combining like terms and reducing complexity. This process often involves consolidating terms with common variables or constants. Consider the expression on the right side of the original problem:

• \(\frac{3}{2}x + \frac{x}{2}\)

Since these terms both have the variable \(x\), they can be combined by adding the coefficients:

• \(\frac{3}{2} + \frac{1}{2} = 2\)

This leads to a simplified form of:

• \(2x\)

Simplifying can also mean removing constants by moving them across the equation. Consistently simplifying expressions during algebraic manipulations aids in clearer, more manageable equations. Essentially, it sets the stage for identifying solutions and understanding the deeper relationships within an equation. The ultimate goal is to make expressions as concise as possible without changing their value.