Problem 27 - Prealgebra

Question

Solve equation. Check your solution. \(3 c+4.5=7.2-6 c\)

Step-by-Step Solution

Answer

The solution is \(c = 0.3\).

Step 1: Move Variable Terms to One Side

To isolate the variable term, let's add \(6c\) to both sides of the equation. This gives us:\[ 3c + 6c + 4.5 = 7.2 \]

Step 2: Combine Like Terms

Combine the \(c\)-terms on the left side:\[ 9c + 4.5 = 7.2 \]

Step 3: Isolate the Variable Term

Now, subtract \(4.5\) from both sides to isolate the term with the variable:\[ 9c = 7.2 - 4.5 \]This simplifies to:\[ 9c = 2.7 \]

Step 4: Solve for the Variable

Divide both sides by \(9\) to solve for \(c\):\[ c = \frac{2.7}{9} \]Simplify the fraction:\[ c = 0.3 \]

Step 5: Check the Solution

Substitute \(c = 0.3\) back into the original equation to verify:\[ 3(0.3) + 4.5 = 7.2 - 6(0.3) \]Calculate each side:Left side: \(0.9 + 4.5 = 5.4\)Right side: \(7.2 - 1.8 = 5.4\)Since both sides are equal, the solution \(c = 0.3\) is correct.

Key Concepts

Algebra

Algebra often revolves around finding the unknown, which is usually represented by a variable like \( c \). This branch of mathematics uses letters to represent numbers in equations, allowing us to solve for unknown values systematically. Understanding algebra involves performing operations such as addition, subtraction, multiplication, and division while maintaining balance on both sides of the equation.
When tackling algebraic equations, you'll frequently encounter the need to rearrange terms and isolate variables. This process helps in simplifying expressions, making the unknown easier to find. Algebra forms the backbone of advanced mathematics, and learning it sets the stage for tackling calculus and other mathematical fields.

Checking Solutions

After finding a solution to an equation, verifying its accuracy is crucial. This process, known as "checking solutions," ensures that the value derived for the variable satisfies the original equation. In our exercise, after finding \( c = 0.3 \), we substitute it back into the original equation:

• The left side becomes \( 3(0.3) + 4.5 = 0.9 + 4.5 = 5.4 \).
• The right side simplifies to \( 7.2 - 6(0.3) = 7.2 - 1.8 = 5.4 \).

Both sides equate to 5.4, confirming the solution is correct. This step is part of ensuring your calculations are error-free, offering peace of mind before you proceed to the next math problem.

Isolating Variables

Isolating the variable is a step in algebra where you aim to get the variable by itself on one side of the equation. This involves moving terms from both sides to achieve a simpler form, helping clarify what value the variable represents. In our exercise, we started with the equation \( 3c + 4.5 = 7.2 - 6c \).
To isolate \( c \), we first added \( 6c \) to both sides, which helped gather all terms with \( c \) on one side. This gave us \( 3c + 6c = 7.2 - 4.5 \), which simplifies further once we combine and simplify the terms. From here, we continued to solve for \( c \), following a clear path that ultimately simplified the equation in a way that made finding \( c \) straightforward.

Combining Like Terms

Combining like terms is an essential skill in solving algebraic equations. It involves simplifying expressions by adding or subtracting terms that have the same variable factor. For example, in our exercise, after moving all the \( c \) terms to one side, we had \( 3c + 6c \).
We then combined these to get \( 9c \), simplifying our equation to \( 9c + 4.5 = 7.2 \). This process helps reduce complexity and brings clarity to equations, making them easier to solve. Mastering the ability to combine like terms allows for a more efficient approach to problem-solving in algebra and lays a solid foundation for tackling more complex equations.