Problem 14 - College Algebra

Question

In Exercises \(1-16,\) solve and check each linear equation. $$ 5 x-(2 x+2)=x+(3 x-5) $$

Step-by-Step Solution

Answer

\(x = 3\)

Step 1: Simplify the Equation

Start by simplifying both sides of the equation independently. On the left side simplify \(5x - (2x+2)\) and on the right side simplify \(x + (3x - 5)\). This results in \(3x - 2 = 4x - 5\).

Step 2: Isolate the Variable Terms

Arrange the equation in such a way that the coefficients of \(x\) are on one side and the constants are on the other side. This can be achieved by subtracting \(3x\) from both sides resulting in \(x = 3\).

Step 3: Validate the Solution

Validate this solution by substituting \(x = 3\) into the original equation \(5x - (2x + 2) = x + (3x - 5)\). If both sides of the equation are equal, then the solution is correct.

Key Concepts

Linear Equation

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations are used widely in various fields. They can represent simple relationships, like the total cost of apples based on the number of kilograms purchased.

In the given exercise, the equation 5x - (2x + 2) = x + (3x - 5) is a straightforward example of a linear equation, which has a single variable, x. The main goal when working with linear equations is to find the value of the variable that makes the equation true. This is often referred to as 'solving' the equation.

Algebraic Manipulation

Algebraic manipulation involves using various algebraic techniques to alter the form of an algebraic expression or equation without changing its value.

These techniques include distributing multiplication over addition, combining like terms, and using properties of equality to maintain the balance of an equation. In our initial step, distributing multiplication over addition involves expanding (2x + 2) and (3x - 5), to integrate these terms with others on their respective sides of the equation. Algebraic manipulation is key in simplifying equations to make solving for the variable more straightforward.

Equation Simplification

Equation simplification is a pivotal part of solving linear equations. This process includes combining like terms and reducing equations to their simplest form, which often makes subsequent steps much easier.

In our equation, simplifying both sides separately resulted in 3x - 2 = 4x - 5. By doing this, we eliminated unnecessary complexity and prepared the equation for the next step, which involved isolating the variable. Simplification may involve adding or subtracting terms on one or both sides, multiplying or dividing, or even factoring. The key is to always strive for the simplest form that accurately represents the original equation.

Variable Isolation

Variable isolation is the ultimate step in solving a linear equation. It involves rearranging the equation in such a way that the variable you're solving for is alone on one side of the equation.

In the solution provided, isolating the variable x was achieved by subtracting 3x from both sides of the simplified equation. This left x on one side and the solution to the equation, 3, on the other side. 3 = 3 is our final check, making sure that the solution is consistent and satisfies the given linear equation. Isolating the variable not only yields the solution but it also provides an easy way to check if the solution is correct by plugging it back into the original equation.