Problem 5 - Prealgebra

Question

Solve equation. Check your solution. \(4 x-1=3 x+2\)

Step-by-Step Solution

Answer

The solution is \(x = 3\). The solution is verified as correct.

Step 1: Isolate Variable Terms

Start by moving all variable terms to one side of the equation. Subtract \(3x\) from both sides:\[4x - 3x - 1 = 3x - 3x + 2\]This simplifies to:\[x - 1 = 2\]

Step 2: Solve for the Variable

Now, isolate the variable \(x\) by adding 1 to both sides of the equation:\[x - 1 + 1 = 2 + 1\]This simplifies to:\[x = 3\]

Step 3: Verify the Solution

Substitute \(x = 3\) back into the original equation to check your answer:\[4(3) - 1 = 3(3) + 2\]Calculate both sides:\[12 - 1 = 9 + 2\]\[11 = 11\]Since both sides are equal, \(x = 3\) is a correct solution.

Key Concepts

Variable Isolation

Variable isolation is the first crucial step when tackling linear equations. It involves rearranging the equation so that the variable you are solving for stands on one side, while the constants and other terms are on the opposite side. A simple way to think about this is to "peel away" layers surrounding the variable much like peeling an onion, till you are left with only the variable.
In our original exercise, like many linear equations, both sides of the equation contain terms with the variable. Our goal is to have all terms containing the variable on one side. To achieve this, you perform operations that maintain equilibrium, similar to what you do with weights on a scale. If you subtract a weight from one side, you must do the same on the other.
Here, subtracting \(3x\) from both sides helps isolate the variable. This steps results in:

• Original: \(4x - 1 = 3x + 2\)
• After Subtraction: \(x - 1 = 2\)

Notice that we are now a step closer to having \(x\) by itself; this sets the stage for solving the equation.

Equation Verification

Equation verification is the final and necessary step in solving equations. It's like double-checking your work to ensure you've landed on the right answer. Verification involves taking the potential solution and plugging it back into the original equation. If both sides of the equation remain equal, your solution is verified as correct.
In our example, once we found that \(x = 3\), we substituted \(3\) back into the original equation to verify. This ensures that the computation was accurate and the operations were properly conducted.

• Substitute: \(4(3) - 1\) and \(3(3) + 2\)
• Calculation: \(11 = 11\)

Both sides equate to \(11\), confirming the solution is correct. This verification provides confidence in your solution and is a good habit to establish for solving any kind of equation.

Prealgebra Concepts

Prealgebra forms the backbone of understanding and solving linear equations. It provides the foundational skills for manipulating numbers and variables. It's about knowing how to deal with basic operations such as addition, subtraction, multiplication, and division while keeping the equation balanced.
During the solution process, we used two critical prealgebra concepts:

• Combining Like Terms: Simplifying \(4x - 3x\) to obtain \(x\)
• Maintaining Balance: Adding \(1\) to both sides simplifies \(x - 1 = 2\) to \(x = 3\)

Understanding these basic concepts allows you to solve equations systematically and effectively. Prealgebra sets the stage for more complex mathematics, ensuring a strong grasp on these principles is key to future success in math.