Problem 52 - College Algebra and Calculus: An Applied Approach

Question

Solve the equation and check your solution. (Some equations have no solution.) $$ 4(x+1)-3 x=x+5 $$

Step-by-Step Solution

Answer

The given equation has no solution since it results in an incorrect statement.

Step 1: Simplify the equation

To simplify the equation, distribute the coefficients into the parentheses. That leads to \(4x + 4 - 3x = x + 5 \). Combine like terms to simplify it further to \(x + 4 = x + 5\)

Step 2: Solve the equation

From the simplified equation \( x + 4 = x+ 5 \), isolate the variable 'x'. But if we subtract 'x' from both sides, we get \( 0 + 4 = 0 + 5 \) or \(4 = 5 \) which is false. Therefore, this equation has no solution.

Step 3: Check the solution

Typically, one would substitute the value found for 'x' in the original equation and check if it is true. But in this case as there is no solution, checking is not applicable.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying mathematical expressions and equations to make them easier to understand or solve. This process involves a variety of techniques such as distributing, combining like terms, and moving terms from one side of the equation to the other using addition or subtraction.

When you encounter an equation like 4(x+1)-3x=x+5, the first step is to distribute any coefficients into the parentheses. This means you'll multiply each term inside the parentheses by the coefficient outside, resulting in an expanded form of the equation. After distributing, combine like terms, which are terms that have the same variable raised to the same power. For example, 4x and -3x are like terms because they both contain the variable x to the first power.

In our exercise, after distributing and combining like terms, the equation simplifies to x + 4 = x + 5. This sets the stage for further analysis to determine whether the equation has a single solution, infinitely many solutions, or no solution at all.

No Solution Equations

Equations with no solution are equations for which no value of the variable will make the equation true. They often result in a false statement, such as a contradiction, once the variable has been eliminated from the equation. This is a vital concept in algebra because it underlines that not all equations have a solvable answer where the variable can be isolated.

For example, when we simplify the provided equation and attempt to solve for x, we reach a point where the variable is eliminated, leaving us with the statement 4 = 5. Since this is a contradiction—it is obviously false and there's no mistake in our manipulation process—it tells us that the original equation has no solution. It's crucial for students to recognize this type of outcome and differentiate it from equations that do have solutions, which would yield true statements when simplified.

Understanding that no value of x can satisfy the equation x + 4 = x + 5 is important for developing problem-solving skills in algebra and for correctly interpreting mathematical models.

Checking Algebraic Solutions

Checking algebraic solutions is the final step in solving equations. It serves to verify that the solution derived is indeed correct and satisfies the original equation. To check a solution, substitute the value of the variable back into the original equation and simplify to see if the equation balances—that is, if the left side equals the right side.

Typically, checking your solution is a straightforward process. However, in the case of our example, where the simplified equation led to a contradiction (4 = 5), there is no solution to check against. It is a critical lesson that checking is not limited to finding a specific numerical solution. Instead, it also includes recognizing when your manipulations lead to a contradiction, indicating that there is no solution.

Students must remember to perform this verification step for every solution they find, as it not only confirms their answer but also helps to catch any potential errors in their algebraic manipulation process. It is a good habit that strengthens mathematical understanding and confidence.