Problem 31 - Algebra Form and Function

Question

Does the equation have no solution, one solution, or an infinite number of solutions? $$ 4 x+3=4(x+1)-1 $$

Step-by-Step Solution

Answer

Answer: The equation has an infinite number of solutions.

Step 1: Distribute 4 on the right side of the equation

Distribute the 4 to both x and 1 on the right side of the equation: $$ 4x + 3 = 4(x+1) - 1 \\ 4x + 3 = 4x + 4 - 1 $$

Step 2: Simplify the equation

Combine the like terms on the right side of the equation: $$ 4x + 3 = 4x + 3 $$

Step 3: Analyze the form of the simplified equation

The equation has simplified as \(4x + 3 = 4x + 3\). Since both sides of the equation are identical, it is a true statement, regardless of the value of x. This means that the equation has an infinite number of solutions.

Key Concepts

Solution Types

In the world of linear equations, understanding the types of solutions you may encounter is crucial. There are typically three kinds of solutions to look out for:

• No solution: This occurs when your equation simplifies to a false statement, like \(3 = 5\). It means there are no values of \(x\) that can satisfy the equation.
• One solution: This happens when the equation simplifies to a clear answer like \(x = 7\). Here, only one specific value for \(x\) makes the equation true.
• Infinite solutions: This appears when both sides of the equation are identical. For example, \(4x + 3 = 4x + 3\) holds true no matter what value \(x\) takes. It implies that every number is a solution.

Recognizing these solution types helps in interpreting equations and understanding their behavior.

Algebraic Manipulation

Algebraic manipulation is a fundamental skill in solving equations. It involves performing operations to both sides of the equation to simplify or solve it. Let's see how it works with the given example:

• Start by distributing any numbers across parentheses. For instance, in our equation \(4(x + 1)\), you multiply both \(x\) and \(1\) by \(4\) to get \(4x + 4\).
• Next, gather like terms to simplify further. In the expression \(4x + 4 - 1\), you can combine \(4\) and \(-1\) to make \(3\).

This step-by-step manipulation helps in revealing the structure of the equation, making it easier to determine the type of solution available.

Infinite Solutions

An equation has infinite solutions when, after simplification, both sides are identical. This was the case in our example:

• After distributing and simplifying, both sides of our equation became \(4x + 3\).
• This means that for any value of \(x\), the equation holds true.

It's crucial to recognize when an equation results in this form as it implies there is not just a single solution, but a whole infinite set. This situation typically arises when you're modeling scenarios where the equation might represent a balance or symmetry that is universally true, irrespective of the specific values involved.