Problem 10 - Elementary Algebra

Question

\(\frac{1}{10} c-\frac{3}{10} c+\frac{7}{10}\)

Step-by-Step Solution

Answer

-\(\frac{1}{5} c + \frac{7}{10})\.

Step 1: Combine Like Terms

Identify and combine the terms with the variable. In this expression, both \(\frac{1}{10} c\) and \(\frac{3}{10} c\) have the variable \c\. Thus, you combine these like terms: \(\frac{1}{10} c - \frac{3}{10} c\).

Step 2: Perform Subtraction

Subtract \(\frac{3}{10} c\) from \(\frac{1}{10} c\): \(\frac{1}{10} c - \frac{3}{10} c = -\frac{2}{10} c\). Simplify the fraction \(-\frac{2}{10}\), which reduces to \(-\frac{1}{5}\). Thus, it becomes \(-\frac{1}{5}c\).

Step 3: Write the Final Expression

After combining and simplifying the terms, the expression simplifies to: \(-\frac{1}{5} c + \frac{7}{10}\).

Key Concepts

Elementary Algebra

Elementary Algebra is the foundation of high school mathematics. It focuses on the manipulation of algebraic expressions and the properties of numbers. In algebra, we often deal with variables (like x and y) and constants (numbers). Understanding basic arithmetic operations like addition, subtraction, multiplication, and division is crucial in solving algebraic problems. These operations are often performed on numbers as well as on variables to form equations and expressions. For example, the exercise involves combining terms involving the variable c, and performing arithmetic operations to simplify the expression.

Combining Like Terms

Combining like terms is an essential step in simplifying algebraic expressions. Like terms are terms that contain the same variables raised to the same power. For example, \(\frac{1}{10} c\) and \(\frac{3}{10} c\) are like terms because they both contain the variable c. To combine them, we add or subtract the coefficients (the numbers in front of the variables). In the given exercise, we subtract \(\frac{3}{10} c\) from \(\frac{1}{10}c\): \(\frac{1}{10} c - \frac{3}{10} c = -\frac{2}{10} c\). Combining like terms reduces the complexity of the expression, making it more manageable.

Fraction Operations

Handling fractions is a critical skill in algebra. Fraction operations include adding, subtracting, multiplying, and dividing fractions. In this exercise, you need to subtract two fractions: \(\frac{1}{10} c\) and \(\frac{3}{10} c\). When subtracting fractions with the same denominator, only the numerators are subtracted. For example, \(\frac{1}{10} c - \frac{3}{10} c = \frac{1-3}{10} c = -\frac{2}{10} c\). After performing the subtraction, it is also crucial to simplify the fraction. Here, \(-\frac{2}{10}\) simplifies to \(-\frac{1}{5}\). Simplifying fractions helps keep your expressions clean and easier to work with.

Expression Simplification

Expression simplification is the process of making an algebraic expression as simple as possible. Simplification involves several steps including combining like terms and simplifying fractions. In the given exercise, we start with the expression \(\frac{1}{10} c - \frac{3}{10} c + \frac{7}{10}\). After combining the like terms and simplifying, you get \(-\frac{1}{5} c\). To write the final expression clearly, we add the constant term \(\frac{7}{10}\): \(-\frac{1}{5} c + \frac{7}{10}\). Simplifying expressions helps in solving equations and understanding the relationship between variables. It makes the math easier to understand and use.