Problem 21 - Prealgebra

Question

Solve each equation. Check your solution. $$(3 x+2)+(-x+5)=2 x-7$$

Step-by-Step Solution

Answer

The equation has no solution.

Step 1: Simplify Both Sides

First, we'll simplify both sides of the equation by combining like terms. On the left side, combine the terms \( 3x \) and \( -x \), resulting in \( 2x + 7 \). Now, the equation looks like this: \[ 2x + 7 = 2x - 7 \]

Step 2: Eliminate Variable from One Side

Subtract \( 2x \) from both sides to eliminate the variable from the left side. This results in:\[ 7 = -7 \]

Step 3: Analyze the Result

Since \( 7 = -7 \) doesn't make sense as a true statement, this means there is no value of \( x \) that will make the original equation true. This indicates that the equation has no solution.

Key Concepts

Combining Like Terms in Algebra

When working with algebraic equations, it's essential to simplify complex expressions by combining like terms. Like terms are terms that have the same variable raised to the same power. For example, in the expression \(3x + 2 - x + 5\), \(3x\) and \(-x\) are like terms because they both contain the variable \(x\). To combine them, simply add or subtract the coefficients: \(3x - x = 2x\). Here's how you can identify and combine like terms:

• Look for terms that have identical variable parts, such as \(3x\) and \(-x\).
• Combine these terms by performing arithmetic operations on their coefficients.
• In this exercise, the expression simplifies to \(2x + 7\) once the like terms are combined.

Combining like terms is a key step in simplifying an equation, paving the way for solving or further manipulating the expression.

No Solution Equations

Sometimes, after simplifying an equation, you might end up with a statement that is always false, such as \(7 = -7\). This occurs in equations that have no solution. When you encounter an equation where the variable cancels out and you are left with a false statement, it means there is no value for the variable that will satisfy the equation.To determine if an equation has no solution:

• Simplify the equation fully by combining like terms.
• Attempt to isolate the variable by performing the same operation on both sides.
• If the variable is eliminated and the remaining statement is false, the equation has no solution.

Having no solution is a specific situation in algebra and is a clear indication that the two sides of the equation cannot ever be equal, no matter what value is substituted for the variable.

Eliminating Variables

Eliminating variables from an equation involves performing operations that simplify the expression by removing the variable from one side. This can often help expose underlying truths about the equation, such as discovering there is no solution.In the exercise, the variable \(x\) is eliminated by subtracting \(2x\) from both sides of the equation, leading to the statement \(7 = -7\). This type of transformation is a powerful tool in algebra because it allows you to see immediately whether the remaining numerical statement is true or false. To eliminate variables effectively:

• Identify terms with the variable you wish to eliminate.
• Perform the same arithmetic operations on both sides of the equation to keep it balanced.
• Check the resulting expression to determine insights about the original problem.

Eliminating variables is not just about simplification—it's about uncovering the consistency or inconsistency within the equation to determine if a valid solution exists.