Problem 31 - Elementary Algebra

Question

Solve. $$ 3 a+5-a=2 a+7 $$

Step-by-Step Solution

Answer

There is no solution, as the equation results in a contradiction.

Step 1: Simplify the Left Side

First, combine like terms on the left-hand side of the equation. We have \(3a + 5 - a\). Combine \(3a\) and \(-a\) to get \(2a\). So the equation becomes \(2a + 5\).

Step 2: Set Up Equation

After simplifying, the equation is now \(2a + 5 = 2a + 7\). Observe both sides of the equation.

Step 3: Eliminate Variable Terms

Subtract \(2a\) from both sides of the equation to eliminate the variable term. The equation will be \(5 = 7\).

Step 4: Analyze the Result

The result \(5 = 7\) is a contradiction, meaning there is no value of \(a\) that will satisfy the equation.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in elementary algebra. It involves finding the value of the variable that makes the equation true. To solve an equation, you usually follow these steps:

• Simplify both sides: Start by simplifying each side of the equation, combining like terms if necessary. This helps present the equation in its simplest form.
• Isolate the variable: Next, try to get the variable on one side of the equation. You can do this by performing operations such as addition, subtraction, multiplication, or division.
• Verify your solution: After finding a solution, it's important to substitute it back into the original equation to ensure that it works.

In our exercise, we first simplified the left-hand side, then attempted to isolate the variable by balancing both sides. The goal is to potentially find a specific solution, or determine the nature of the solution set. In this case, due to a contradiction, no solution exists.

Combining Like Terms

Combining like terms is an essential part of simplifying algebraic expressions. Like terms are terms that have identical variable parts. For example, in the expression \(3a + 5 - a\), both \(3a\) and \(-a\) are like terms because they both contain the variable \(a\). Here’s how you combine them:

• Identify like terms: Look for terms with the same variables raised to the same power.
• Combine coefficients: Add or subtract the coefficients of the like terms while keeping the variable part unchanged. For instance, \(3a - a\) becomes \(2a\).
• Simplify further if needed: Once combined, your expression should be easier to manage. Continue simplifying the rest of the expression.

In the exercise, identifying and combining the like terms \(3a\) and \(-a\) gave us \(2a\), simplifying the equation and making it easier to solve.

Contradiction in Equations

Sometimes, while simplifying equations, you might end up with a statement that is clearly false, like \(5 = 7\). This indicates a contradiction, meaning there is no solution that satisfies the equation. Here’s how to understand this situation:

• Recognize contradicted equations: When simplifying both sides leads to obviously false statements, it shows that the equation has no solution.
• Understand what it signifies: A contradiction often occurs when the problem was set up initially with incompatible conditions or operations that eliminate the variable completely.

This exercise led to the contradiction \(5 = 7\), indicating the original equation \(3a + 5 - a = 2a + 7\) has no valid solution. This calls attention to analyzing problems meticulously, as not all equations have solutions.