Problem 32 - 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 no solution.

Step 1: Write down the given equation

We are given the equation: $$ 4x + 3 = 4(x+1) + 1 $$

Step 2: Distribute the 4 on the right-hand side

We distribute the 4 among the terms within the parentheses on the right-hand side: $$ 4x + 3 = 4x + 4 + 1 $$

Step 3: Simplify the right-hand side

Combine the constant terms on the right-hand side: $$ 4x + 3 = 4x + 5 $$

Step 4: Analyze the simplified equation

We notice that both sides of the equation have \(4x\). If we subtract \(4x\) from both sides, we get the following: $$ 3 = 5 $$ Since 3 and 5 are different constants, this equation is false, showing that there is an inconsistency within the original equation.

Step 5: Conclusion

Due to the inconsistency, the original equation has no solution.

Key Concepts

Equation Simplification

Simplifying an equation means reducing it to its simplest form where further calculations are easier to perform. It's like cleaning up a messy room to see everything clearly.
The goal is to gather all similar terms together. You do this by combining terms, like adding or subtracting constants or variable terms from both sides.
In our original equation, after distributing, we had:

• The left side: \(4x + 3\)
• The right side: \(4x + 4 + 1\)

Notice that on the right-hand side, \(4x\) is already simplified, but the constants \(4\) and \(1\) add up to \(5\).
This means our equation simplifies to \(4x + 3 = 4x + 5\). Each expression on either side is now as straightforward as possible.

Distributive Property

The distributive property is a vital tool in algebra, allowing us to multiply a single term by terms inside a parenthesis. It helps simplify an equation while ensuring each term within the parentheses is properly accounted for.
In simple terms, to distribute means to "pass out" or "spread" the multiplier to all elements within the parentheses. Mathematically, if \(a(b + c)\), then \(ab + ac\).
In the given equation \(4(x + 1) + 1\), the term \(4(x+1)\) uses the distributive property. We multiply \(4\) with each term in \((x + 1)\), turning it into \(4x + 4\).
This moves us closer to simplifying, and eventually solving, the equation.

Consistent and Inconsistent Equations

Equations can be classified based on the solutions they offer. Understanding the terms consistent and inconsistent equations helps in knowing what to expect from the results.
A **consistent equation** has at least one valid solution. This means there's a number that, when substituted for the variable, makes the equation true. Depending on the equation, it might have one, or even infinitely many solutions for linear equations.
On the other hand, an **inconsistent equation** offers no solution. When you simplify such an equation, you end up with a statement that is logically false, such as \(3 = 5\), as in our simplified equation.
This inconsistency indicates there's no value you can substitute for \(x\) to make the equation true. Recognizing this early saves time and tells us that the original setup of the equation needs re-evaluation.