Problem 65 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation. Use words or set notation to identify equations that have no solution, or equations that are true for all real numbers. \(4 x+1-5 x=5-(x+4)\)

Step-by-Step Solution

Answer

The given equation is true for all real numbers, meaning that it has infinitely many solutions.

Step 1: Simplify both sides

Starting with the given equation, \(4x + 1 - 5x = 5 - (x + 4)\), simplify both sides of the equality by making use of the commutative and associative properties of addition. The left side simplifies to \(-x + 1\), and the right side simplifies to \(5 - x - 4\). This results in the equation \(-x + 1 = 5 - x - 4\).

Step 2: Collect similar terms

The next step is to collect similar terms. The equation becomes \(-x + 1 = -x + 1\). This resulted after simplifying \(5 - x - 4\) on the right side.

Step 3: Compare both sides

Subtract \(-x\) and 1 from both sides of the equation to maintain equality. After doing this, we get \(0 = 0\).

Step 4: Interpret the equation

The equation \(0 = 0\) is always true, for every real number. This means there are infinitely many solutions, and every real number is a solution to the initial equation. This situation occurs when the two expressions on either side of the equation are equivalent for all values of \(x\).

Key Concepts

Equation Solving Steps

Mastering the art of solving algebraic equations opens up a world of possibilities in mathematics and beyond. Begin by examining the given problem: simplify each side of the equation, group like terms, and use inverse operations to isolate the variable.

For the given exercise, we first simplified each side, which is our first step. This involves combining terms and removing parentheses. In the next step, we grouped like terms together to further streamline the equation. Finally, we worked to isolate the variable. However, in this case, the variable was eliminated, leading us to an interesting conclusion which is explained in the subsequent sections.

Properties of Equality

The properties of equality are fundamental tools that allow us to manipulate equations confidently. These properties include the reflexive property (\(a = a\)), symmetric property (\(a = b\rightarrow b = a\)), transitive property (\(a = b\text{ and } b = c\rightarrow a = c\)), addition and subtraction properties (\(a = b\rightarrow a \underline{\phantom{xxx}} \bSc{\text{±}} \underline{\phantom{xxx}} c = b \underline{\phantom{xxx}} \bSc{\text{±}} \underline{\phantom{xxx}} c\)), and multiplication and division properties (\(a = b\text{ and }ceq0\rightarrow a \underline{\phantom{xxx}} \bSc{\text{×}} \underline{\phantom{xxx}} (c) = b \underline{\phantom{xxx}} \bSc{\text{×}} \underline{\phantom{xxx}} (c)\)).

It's important to apply these properties with precision to ensure each algebraic operation we perform is justified. In our exercise, we primarily used the addition property to subtract terms from both sides, maintaining the balance of the equation—fundamental to the concept of equality.

No Solution Equations

There are scenarios where an equation can be simplified to a point where it leads to a contradiction such as \(3 = 4\), which indicates that no solution exists because there is no possible value of the variable that can satisfy the equation. However, our initial equation simplifies to \(0 = 0\), which is always true, and this is not a contradiction but an identity, leading to a different conclusion called infinitely many solutions, or an identity equation.

Identities in Algebra

Identities are special equations that are true for all values of the variables they contain. In this case, once simplified, our exercise resulted in \(0 = 0\), which is an identity since it holds true regardless of the value of \(x\).

Identifying an identity is crucial because it tells you that the original equation is true for all real numbers. It is a fundamental concept that has a vast number of applications in algebra, and understanding how to recognize an identity can significantly enhance problem-solving abilities.