Problem 58 - Algebra and Trigonometry with Analytic Geometry

Question

Determine which equation is not equivalent to the equation preceding it. $$ \begin{aligned} 5 x+6 &=4 x+3 \\ x^{2}+5 x+6 &=x^{2}+4 x+3 \\ (x+2)(x+3) &=(x+1)(x+3) \\ x+2 &=x+1 \\ 2 &=1 \end{aligned} $$

Step-by-Step Solution

Answer

The fourth equation \(x + 2 = x + 1\) is not equivalent to the preceding equations.

Step 1: Analyzing the first equation

Consider the equation \(5x + 6 = 4x + 3\). Subtract \(4x\) from both sides to isolate \(x\) on one side, resulting in \(x + 6 = 3\). Next, subtract \(6\) from both sides to solve for \(x\), giving \(x = -3\). This tells us that \(x = -3\) is a solution to the equation.

Step 2: Analyzing the second equation

The equation is \(x^2 + 5x + 6 = x^2 + 4x + 3\). Subtract \(x^2 + 4x + 3\) from both sides, simplifying to \(x + 3 = 0\), which solves to \(x = -3\) again, consistent with our first equation.

Step 3: Analyzing the third equation

The equation \((x+2)(x+3) = (x+1)(x+3)\) should be expanded. Expand both sides with the distributive property: on the left-hand side, it becomes \(x^2 + 5x + 6\), and on the right-hand side, \(x^2 + 4x + 3\). This is identical to the second equation, so it similarly simplifies to \(x = -3\).

Step 4: Analyzing the fourth equation

The equation \(x + 2 = x + 1\) can be simplified by subtracting \(x\) from both sides, resulting in \(2 = 1\). Since \(2\) does not equal \(1\), this equation does not hold true and is not equivalent to any of the preceding equations.

Step 5: Analyzing the fifth equation

The equation \(2 = 1\) is a direct inconsistency. It confirms that the solution of the fourth equation does not hold true, reinforcing that the issue lies with the fourth equation.

Key Concepts

Step-by-Step Solution

Solving equivalence problems in algebra often requires detailed analysis, where each step is broken down methodically. These steps are designed to highlight how equations can be transformed while still representing the same solution. Here, each equation builds upon the last, until an inconsistency is found. The process involves rearranging algebraic expressions, simplifying terms, and verifying solutions at each stage. This meticulous approach aids in understanding how equations relate to each other and helps identify any discrepancies.

Algebraic Manipulation

Algebraic manipulation involves transforming equations through valid operations such as addition, subtraction, multiplication, and division. This allows us to isolate variables and simplify expressions. In the given exercise, manipulation begins with isolating terms involving \(x\). For example, subtracting \(4x\) from both sides of the equation \(5x + 6 = 4x + 3\) gives us \(x + 6 = 3\). Subsequent subtraction of \(6\) isolates \(x\), showing that \(x = -3\). These steps help to explore equivalent forms and ensure each equation still represents the same underlying relationship.

Equation Solving

Equation solving is a fundamental part of algebra. It involves finding the value of unknowns that satisfy the equation. In this exercise, each equation from the sequence is solved separately to check if they represent the same solution. Solving is carried out by simplifying expressions and checking solutions across equations. As seen in the third equation, expanding

• \((x+2)(x+3)\)
• \((x+1)(x+3)\)

shows a consistency with previous solutions. This confirms the validity of steps and helps in identifying the non-equivalent equation.

Inconsistency

In mathematics, an inconsistency occurs when an equation results in contradictory outcomes, such as \(2 = 1\). In the exercise, analyzing the fourth and fifth equations reveals that they lead to such inconsistencies. By simplifying the equation \(x + 2 = x + 1\), subtracting \(x\) gives \(2 = 1\), which is clearly not true. This contradiction indicates that the fourth equation is distinct from others. Identifying these inconsistencies is key in recognizing errors or outliers in algebraic sequences.

Algebra

Algebra is the cornerstone of solving and understanding mathematical equations that involve variables. It lays out methods for manipulating these symbols to solve for unknowns. Across the exercise, algebra serves as the framework for handling expressions and performing operations that test equivalence. By applying fundamental principles, such as the distributive property in expanding expressions or isolating terms to solve equations, one can thoroughly analyze the relationships within a sequence of problems to verify their equivalence or identify discrepancies.