Problem 51 - Algebra A Combined Function

Question

Solve each equation. $$ \begin{aligned} &3 x-4=3(x+1)\\\ \end{aligned} $$

Step-by-Step Solution

Answer

The equation has no solution.

Step 1 - Distribute the Right Side

First, apply the distributive property to the right side of the equation. You'll distribute the 3 across the terms inside the parentheses: \( 3(x+1) = 3\cdot x + 3\cdot 1 = 3x + 3 \). Now rewrite the equation as: \[ 3x - 4 = 3x + 3 \].

Step 2 - Eliminate the Variable on One Side

Subtract \(3x\) from both sides of the equation to eliminate the \(3x\) term: \( 3x - 4 - 3x = 3x + 3 - 3x \). This simplifies to: \[ -4 = 3 \].

Step 3 - Analyze Result for No Solution

Notice that the resulting simplified equation \( -4 = 3 \) is not true. This indicates that there is a contradiction, meaning the original equation has no solution.

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows you to multiply a single term by two or more terms inside a parenthesis. It's helpful for simplifying expressions and solving equations. To understand this property better, let's look at the example given in the problem:

We start with the expression on the right side:

• 3 multiplied by each term inside the parenthesis (x + 1).
• This results in 3x + 3.

This property distributes the multiplication over addition. It's crucial because it lets you deal with terms separately. You'll often use it to remove parentheses, making equations easier to work with.

No Solution Equations

In algebra, sometimes you encounter equations that have no solution. These are known as "no solution" equations, and they occur when the statement you derive is false, like in this example.

After applying the distributive property and simplifying both sides, the equation:

• turns into -4 = 3.
• This is clearly a contradiction because -4 can never equal 3.

When you see such contradictions after simplifying, it signals the equation has no solution. It means there isn't a value for the variable that would make the equation true. This insight helps to quickly spot and understand such scenarios in algebraic problems.

Variable Elimination

Variable elimination is a critical part of solving equations, especially when determining if an equation has a solution. In the given problem, strapping the variable to one side reveals much about the equation's nature.

We see this when:

• You subtract 3x from both sides to isolate the variable.
• This results in the equation -4 = 3.

By doing this, you cancel out or eliminate the variable, allowing you to focus on the constants or numbers left. This step is invaluable for checking the validity of equations as it simplifies the equation further. To verify the result accurately, make sure to perform the elimination step systematically on both sides of the equation. This keeps the equation balanced and confirms whether you reached a truth (equality) or contradiction.