Problem 32 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The equation is a contradiction; no solution exists.

Step 1: Simplify both sides

Start by expanding and simplifying the equation given: \(-3x = -2x + 1 - (5 + x)\). Distribute the negative sign through the terms inside the parentheses:\(-3x = -2x + 1 - 5 - x\).

Step 2: Combine like terms

Combine the like terms on the right side of the equation. This gives us:\(-3x = -2x - x + 1 - 5\).Simplify it further to:\(-3x = -3x - 4\).

Step 3: Analyze the resulting equation

Now, analyze the simplified equation:\(-3x = -3x - 4\). Notice that when you add \(3x\) to both sides, the equation becomes:\(0 = -4\).

Step 4: Determine if the equation is an identity or contradiction

Since \(0 = -4\) is a false statement, this tells us that no value of \(x\) will satisfy the original equation, making it a contradiction.

Key Concepts

Equation Solving

Solving an equation in algebra involves finding the value of the unknown variable that makes the equation true. To perform equation solving effectively, we need to systematically simplify both sides of the equation. This usually means:

• Removing parentheses by distributing any numbers or negative signs outside them.
• Combining "like terms" to make the equation easier to work with.
• Balancing both sides by using addition, subtraction, multiplication, or division.

In the provided exercise, the initial step involved expanding the equation to eliminate the parentheses, thus paving the way for clearer computation. By making both sides of the equation simpler, we arrive at an expression that is more straightforward to analyze.

Identities and Contradictions

In algebra, equations can have different types of solutions, including identities and contradictions. An identity occurs when both sides of the equation are identical for all values of the variable, resulting in an infinite number of solutions. A classic example is when simplifying leads you to something like \( x = x \); this means any value for \( x \) will satisfy the equation.
On the other hand, a contradiction means there are no solutions to the equation. This happens when, after simplification, you end up with a false statement, such as\( 0 = -4 \).
In our exercise, simplifying resulted in a contradiction, indicating no value of \( x \) could ever satisfy the given equation. This insight is essential in identifying whether a further search for solutions is necessary or not.

Like Terms

"Like terms" in algebra are terms that have the same variables raised to the same powers. Combining like terms is a crucial step in simplifying equations, making the process of solving them more manageable.
For example, in the expression \( -2x - x \), both terms are like terms because they both contain the variable \( x \) raised to the same power. Combining them results in \( -3x \). This operation simplifies the equation and makes it easier to solve.

• Always look to combine terms that have the exact same letters and powers.
• Remember to also consider the coefficients and signs in front of the terms.

In the exercise, combining like terms on the right side of the equation led to a much clearer picture of what was happening in the equation.