Problem 48 - Introductory Algebra for College Students

Question

Solve each equation and check your proposed solution in Exercises. $$1.2 x-3.6=2.4-0.3 x$$

Step-by-Step Solution

Answer

The solution to the equation \(1.2 x - 3.6 = 2.4 - 0.3 x\) is \(x = 4\).

Step 1: Simplify the Equation

First, isolate the \(x\) terms on one side of the equation and the constant terms on the other side. To do this, add \(0.3x\) to both sides of the equation and add \(3.6\) to both sides. This results in the equation: \(1.5 x = 6.0\)

Step 2: Solve for \(x\)

To isolate \(x\), you have to divide both sides of the equation by the coefficient of \(x\) which is \(1.5\). So, \(x = \frac{6.0}{1.5} = 4\). So, \(x = 4\) is the solution.

Step 3: Check Proposed Solution

To verify this solution, substitute \(x = 4\) into the original equation: \(1.2(4) - 3.6 = 2.4 - 0.3(4)\). Simplify each side: \(0.0 = 0.0\). As the left side equals the right side, the proposed solution is correct.

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is the first step in solving linear equations. In essence, an algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \( x \)), and operators (such as add, subtract, multiply, and divide). The beauty of algebraic expressions lies in their ability to represent real-life problems in a simplified numerical way.

For example, in the given expression \(1.2x - 3.6\), \(1.2x\) and \(-3.6\) are termed as 'terms' of the expression. The variable term \(1.2x\) indicates 1.2 times a certain number which we don't know yet, and the constant term \(-3.6\) is a specific number that subtracts from the variable term. Recognizing each part of the expression is essential for manipulating and ultimately solving the equation.

Equation Simplification

Once you're comfortable with recognizing algebraic expressions, the next step is simplifying the equation. Simplification may involve combining like terms, which are terms that have the same variables raised to the same power, and managing constants to move closer to the solution.

Here are some key strategies for simplification:

• Moving all variable terms to one side of the equation and constants to the other side.
• Using inverse operations to simplify the equation. For example, if a term is subtracted, you would add the same value to both sides to cancel it out.
• Making the coefficient of the variable term a '1' to isolate the variable, if it isn't already.

For instance, in the step-by-step solution, combining \(1.2x\) and \(0.3x\) resulted in \(1.5x\), and moving the constant \(-3.6\) to the other side of the equation simplified it to \(6.0\). The equation is now in a form that is easier to solve.

Solution Verification

The last and crucial phase in solving linear equations is to verify the solution. Don't ever skip this step! Verification ensures that the solution you obtained is indeed correct, reinforcing your understanding and confidence in algebra. To verify a solution, simply substitute the value of the variable back into the original equation and ensure that the resulting expression holds true.

For example, after solving the given equation, the value of \(x\) was found to be 4. To verify this, plug \(x = 4\) back into the original equation: \(1.2(4) - 3.6 = 2.4 - 0.3(4)\). Simplifying each side should yield an identity, such as \(0.0 = 0.0\), which confirms that \(x = 4\) is the correct solution. This process not only proves your solution is right but also provides a solid understanding of the relationship between the algebraic expressions and the operations carried out to solve the equation.