Problem 69 - Introductory Algebra for College Students

Question

Solve each equation. Use words or set notation to identify equations that have no solution, or equations that are true for all real numbers. $$3-x=2 x+3$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0\).

Step 1: Simplify the equation

To simplify the equation \(3 - x = 2x + 3\), we can gather like terms together by adding \(x\) to each side and subtracting \(3\) from each side. This gives us \(3 - x + x - 3 = 2x + x + 3 - 3\) and simplifies to \(0 = 3x\)

Step 2: Solve for the variable

To solve for \(x\), we can divide each side of the equation by 3. This gives us \(0 / 3 = 3x / 3\) which simplifies to \(0 = x\).

Step 3: Check for solutions

Substitute \(x = 0\) back into the original equation \(3 - x = 2x + 3\) to double check if there is a solution. The equation becomes \(3 - 0 = 2*0 + 3\) which simplifies to \(3 = 3\), which is true. Therefore, the equation has solution.

Key Concepts

Algebraic Equations

At the heart of algebra are algebraic equations, which are mathematical statements indicating that two expressions are equal. They involve variables, which are symbols used to represent unknown values that we aim to find. A basic example of an algebraic equation is the linear equation, which has the general form of ax + b = 0, where a and b are constants, and x is the variable.

In the given problem 3 - x = 2x + 3, we are looking for a value of x that makes the equation true. Solving this equation involves finding the value of the variable that balances the equation, showing the concept of equality in algebra. This type of problem-solving is fundamental because it builds the foundation for more complex mathematical concepts and applications.

Equation Simplification

The simplification of equations is an essential skill in solving them. Simplifying an equation means to manipulate it in such a way that it becomes easier to solve, typically resulting in a simpler form. The key steps often involve combining like terms and using inverse operations.

In our exercise, we initially have an equation where the variable x appears on both sides. The simplification process involves adding x to and subtracting 3 from both sides, which results in 0 = 3x. This is a straightforward equation that indicates that the product of 3 and the variable equals 0, steering us towards the solution. In simplifying equations, one must always perform the same operation on both sides to maintain balance, which is the essence of the equality principle.

Set Notation in Math

Set notation is a way of representing a well-defined collection of objects, which could be numbers, characters, or other entities. In mathematics, sets are typically noted using curly braces, such as {1, 2, 3} or using interval notation for a range of numbers, like (0, ∞) for all positive numbers.

When identifying solutions to an equation, set notation can express the solution concisely. For example, if an equation has only one solution, like our exercise where x = 0, we could write the solution set as {0}. In more complex scenarios where there may be multiple or no solutions, set notation provides a clear and structured way to convey this information.

No Solution and Identity Equations

Understanding that some equations have no solution, while others are true for all values of the variable, is crucial. An equation has 'no solution' when no value of the variable can satisfy it. This situation is noted by writing ∅, which represents the empty set. In contrast, an identity equation is always true, regardless of the value of the variable. Such equations are expressed as identities, like x = x, and have infinite solutions represented by {all real numbers} or (-∞, ∞).

Our exercise led to a specific solution, x = 0, indicating it is neither an identity nor a 'no solution' equation. However, in problems where rearranging and simplifying lead to a false statement like 0 = 5, we would conclude there's no solution; or an equality like 0 = 0, where it's true for all real numbers, signaling an identity equation.