Problem 76 - Algebra A Combined Function

Question

Match each equation in the first column with its solution in the second column. Items in the second column may be used more than once. a. all real numbers b. no solution C. 0 $$ 3 x+1=3 x+2 $$

Step-by-Step Solution

Answer

No solution.

Step 1: Identify the Equation

We have the equation \(3x + 1 = 3x + 2\) that we need to solve.

Step 2: Subtract \(3x\) from Both Sides

Subtract \(3x\) from both sides of the equation to eliminate the \(3x\) terms: \[ 3x + 1 - 3x = 3x + 2 - 3x \]This simplifies to: \[ 1 = 2 \]

Step 3: Analyze the Result

Notice that the resulting equation \(1 = 2\) is a contradiction, which means it is never true.

Step 4: Conclude the Solution

Since the equation \(1 = 2\) is always false, there is no possible value of \(x\) that can satisfy the original equation \(3x + 1 = 3x + 2\). Therefore, the solution is 'no solution'.

Key Concepts

Solving Linear Equations

Linear equations are mathematical expressions involving variables, constants, and operations like addition, subtraction, multiplication, and division. The goal when solving these equations is to find the value of the variable that makes the equation true. Let's use an example equation to illustrate this: suppose we have the equation \(2x + 3 = 7\). To solve it, we need to isolate \(x\) on one side of the equation.

• First, subtract 3 from both sides to get: \(2x = 4\).
• Next, divide both sides by 2 to solve for \(x\): \(x = 2\).

In this process, we're essentially undoing the operations around \(x\) to get it by itself. The solution \(x = 2\) is the value that satisfies the original equation.
When solving linear equations, the steps involve ensuring that every operation you perform keeps the equation balanced. This is crucial for obtaining the correct solution.

Contradiction in Equations

Sometimes, you'll encounter equations that seem impossible to solve because they lead to statements that are inherently false, known as contradictions. In algebra, a contradiction occurs when simplification of the equation results in a statement like \(1 = 2\).
To identify a contradiction:

• Isolate terms involving variables on one side of the equation.
• Perform operations to simplify the equation as much as possible.
• If you end up with a contradictory statement (e.g., \(5 = 3\)), the equation has something wrong with it in terms of having a valid solution.

The presence of a contradiction in an equation indicates that it's structured in a way that no value for the variable will make it true, which brings us to the next concept: no solution.

No Solution in Algebra

In algebra, there are instances where an equation has no solution. This typically occurs when, after simplifying the equation, you are left with a contradiction. In our original example, \(3x + 1 = 3x + 2\), simplifying leads to \(1 = 2\), a contradiction.

• When faced with such contradictions, the equation essentially has no solution, because there's no value for the variable that will satisfy the equation.
• This result is often indicated by saying the solution set is 'empty' or writing it as \(\emptyset\).

Recognizing when an equation like this has no solution is important in algebra, as it helps avoid spending time looking for an answer that does not exist. Always double-check the work to ensure the equation has been simplified correctly before concluding there is no solution.