Problem 73 - Intermediate Algebra

Question

Solve each equation. If the equation is an identity or a contradiction, so indicate. See Example 10. $$ -3 x=-2 x+1-(5+x) $$

Step-by-Step Solution

Answer

The equation is a contradiction.

Step 1: Simplify the Equation

Start by distributing and combining like terms on the right side of the equation. The equation is:\[-3x = -2x + 1 - (5 + x)\]First, distribute the negative sign across the parentheses:\[-3x = -2x + 1 - 5 - x\]Then, combine the constants on the right side:\[-3x = -2x - x + 1 - 5\]This simplifies to:\[-3x = -3x - 4\]

Step 2: Bring Variables to One Side

To simplify further, add \(3x\) to both sides of the equation to start isolating terms:\[-3x + 3x = -3x + 3x - 4\]This results in:\[0 = -4\]

Step 3: Analyze the Result

The equation \[0 = -4\] is a contradiction, as it suggests a false statement (0 can never equal -4). Therefore, the original equation is a contradiction.

Key Concepts

Identity Explained

In algebra, an *identity* is an equation that is true for all values of the variables involved. This means no matter what number is substituted for the variable, the equation holds true. Consider an equation like \( x + 2 = x + 2 \). Whether \( x \) is 5, -1, or a million, both sides of the equation remain equal. Hence, such equations express a universal truth.When solving for an algebraic equation, if all variable terms cancel out and a true result like \( 0 = 0 \) remains, the equation is considered an identity. It's as if both sides of the scale are perfectly balanced for every possible weight. Recognizing identities is crucial in algebra because they inform us that the equation doesn’t just hold for some peculiar values, but universally so for any number substituted in place of the variables.

Understanding a Contradiction

In contrast to an identity, a *contradiction* is an equation that is never true. For any value of the variable, the equation fails to balance. The example given, \( -3x = -2x + 1 - (5 + x) \) simplifies, upon proper manipulation, to a result like \( 0 = -4 \). This final equation is impossible in the real number system.Here’s why:

• When solving equations, if all the variable terms cancel, leaving behind a contradictory result like \( 0 = -4 \), it's a sign you’ve reached a contradiction.
• No feasible number exists that could balance this statement since 0 is never going to equal -4.

Contradictions tell us that the original equation has no solution. Recognizing such outcomes is key because it saves effort in searching for solutions where there are none. It's akin to searching for a solution to a mystery that doesn't have a resolution.

The Art of Equation Simplification

Equation simplification is a fundamental skill in algebra that involves reducing an equation to its simplest form. This process helps both in solving the equation and in understanding its nature, whether it's an identity, a contradiction, or neither. Here are some basic steps:

• Distribute: Apply the distributive property, like removing parentheses by distributing multiplication over addition or subtraction.
• Combine like terms: Merge terms that have the same variables raised to the same power, making the equation simpler.
• Isolate the variable: Move terms with variables to one side and constant terms to the other.

In the given exercise, starting with distribution and then combining like terms simplified the equation enough to discern whether it held true universally, was never true, or depended on specific conditions. Simplification reveals the underlying form of an equation, much like peeling away the layers of an onion to get to its core. By consistently practicing these steps, algebraic problems become more approachable and clearer to solve.