Problem 92 - Algebra A Combined Function

Question

The equation \(6 x+2-2 x=4 x+1\) has no solution. Try solving this equation for \(x\) and describe what happens.

Step-by-Step Solution

Answer

The equation is a contradiction; it has no solution as shown by the false statement 2 = 1.

Step 1: Simplify Both Sides

Start by simplifying both sides of the equation. On the left side, combine like terms: \(6x - 2x + 2 = 4x + 2\). The right side remains the same, \(4x + 1\).

Step 2: Rearrange the Equation

Move all terms involving \(x\) to one side to see if you can isolate \(x\). Subtract \(4x\) from both sides: \(6x - 2x - 4x + 2 = 1\). This simplifies to \(0\cdot x + 2 = 1\).

Step 3: Analyze the Result

You get \(2 = 1\), which is a contradiction. There are no values of \(x\) that can make this equation true, indicating that the original equation has no solution.

Key Concepts

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. A linear equation is an equation of the first degree, which means it has no exponents greater than one. To solve these equations, follow these simple steps:

• First, simplify both sides of the equation if necessary. This means you should combine any like terms and make the equation as simple as possible.
• Next, you should try to get all the terms with the variable on one side and the constants on the other side. This often involves adding or subtracting terms from both sides.
• Finally, isolate the variable by dividing or multiplying both sides of the equation so that you have the variable by itself.

If you follow these steps, you can find the value of the variable that makes the equation true. However, sometimes you may encounter equations that cannot be solved, or have no solution, as we will discuss next.

Equations with No Solution

Sometimes, when solving linear equations, you might end up with a statement that is always false, such as \(2 = 1\). If your work leads to a false statement like this, it means the equation has no solution. This happens because there is no value for the variable that can make the equation true under standard mathematics.
Equations with no solution occur when the two sides of the equation are completely different no matter the value of the variable, often due to cancellation or simplification errors. Recognizing equations with no solution is crucial as it saves you time from trying to force a solution where none exists. Always double-check the initial equation setup for any simplification mistakes that might have led to the false statement to ensure it's indeed a no-solution scenario.

Like Terms

Like terms in an equation are terms that contain the same variables raised to the same power. For example, in the expression \(6x - 2x\), both terms are like terms because they contain the variable \(x\) raised to the same power.
To simplify an expression, you can combine like terms by adding or subtracting their coefficients. This helps by reducing the number of terms in an equation, making it easier to solve.

• For example, if you have \(6x - 2x + 2\), combine \(6x\) and \(-2x\) to get \(4x\), thus simplifying the expression to \(4x + 2\).
• Simplification through like terms helps in solving equations faster by making them less cumbersome.

Remember, being able to identify and combine like terms is foundational in successfully solving algebraic equations.

Isolating Variables

Isolating the variable is a critical step in solving any algebra equation. This means getting your variable, generally \(x\), on one side of the equation without any accompanying numbers or terms.
Here are the common strategies used to isolate a variable:

• Perform operations that reverse or "undo" operations surrounding the variable. For instance, if you have \(4x = 8\), you would divide both sides by 4 to isolate \(x\), giving you \(x = 2\).
• Move terms to the other side of the equation by adding or subtracting. For instance, if you have \(x + 5 = 10\), subtract 5 from both sides to isolate \(x\), resulting in \(x = 5\).
• Always perform the same operation on both sides of the equation to maintain balance.

Being proficient at isolating variables will aid you in efficiently solving algebraic problems and is a useful strategy across various mathematical disciplines.