Problem 36 - Elementary Algebra

Question

Solve. $$ 3 x-0.75+0.21 x=1.24 x+7.13 $$

Step-by-Step Solution

Answer

x = 4.

Step 1: Combine Like Terms

Start by identifying and combining like terms. The original equation is \(3x - 0.75 + 0.21x = 1.24x + 7.13\). Combine \(3x\) and \(0.21x\) on the left-hand side of the equation to get \(3.21x - 0.75\). Now the equation becomes \(3.21x - 0.75 = 1.24x + 7.13\).

Step 2: Move Variable Terms to One Side

Subtract \(1.24x\) from both sides of the equation to get the variable terms on one side. The equation now becomes \(3.21x - 1.24x - 0.75 = 7.13\), simplifying to \(1.97x - 0.75 = 7.13\).

Step 3: Isolate the Variable Term

Add \(0.75\) to both sides to isolate the variable term \(1.97x\). The equation becomes \(1.97x = 7.88\).

Step 4: Solve for x

Divide both sides of the equation by \(1.97\) to solve for \(x\). This gives \(x = \frac{7.88}{1.97}\).

Step 5: Calculate the Solution

Perform the division \(7.88 \div 1.97\) to find the value of \(x\). The result is \(x = 4\).

Key Concepts

Combining Like Terms

When solving linear equations, combining like terms is a fundamental skill. This involves identifying terms in the expression that have the same variable raised to the same power. For example, consider the expression \(3x + 0.21x\). Both terms contain \(x\) as the variable. Therefore, they are like terms and can be added together to simplify the equation. The combined result is \(3.21x\).

Combining like terms is crucial because it reduces the complexity of the equation, making it simpler to solve. It ensures that you work with a streamlined equation, which is easier to manipulate when moving to the next steps in solving the equation. Remember always to perform operations such as addition or subtraction as indicated, to consolidate these terms effectively.

Isolation of Variables

Isolation of variables refers to the process of rewriting the equation so that the variable is on one side, and all constants are on the other. This is crucial because it directly leads to finding the value of the variable. In our case, from the equation \(1.97x - 0.75 = 7.13\), we added \(0.75\) to both sides to isolate the term \(1.97x\).

This step requires careful manipulation of the equation. You perform operations that "undo" additions or subtractions involving the variable's coefficient on one side. By systematically performing the same operation on both sides, you maintain the equation's balance and slowly peel away layers surrounding the variable term.

Solving Equations

After isolating the variable term, our task is to solve for the variable by making it the subject of the formula. The simplified equation "\(1.97x = 7.88\)" shows \(1.97\) as the coefficient of \(x\). To solve for \(x\), divide both sides of the equation by \(1.97\). This operation effectively simplifies the problem to finding \(x\) on its own, resulting in \(x = \frac{7.88}{1.97}\).

Solving equations often involves inverse operations. If the variable is multiplied by a number, you divide both sides by that number. If it is divided, you multiply by the divisor. Remember, consistency is key—perform every operation to both sides to keep things equal.

Step-by-Step Solution

A step-by-step solution approach helps in understanding each part of problem-solving distinctly and thoroughly. Breaking down complex problems into smaller, manageable stages allows for easier comprehension and error detection. Let's review our steps:

• Combine like terms to simplify the equation initially.
• Move all terms involving the variable to one side, and constants to the other.
• Isolate the variable by removing surrounding constants.
• Divide by the coefficient of the variable to finally solve for it.
• Perform the math operations at each stage accurately to ensure a correct solution.

A clear, methodical process fosters a deeper understanding and presents a logical path to follow when tackling similar problems.