Problem 12 - Algebra A Combined Function

Question

$$ \left(-\frac{4}{7} n^{2}+\frac{5}{6} m-\frac{1}{20}\right)+\left(\frac{3}{7} n^{2}-\frac{5}{12} m-\frac{3}{10}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{1}{7} n^{2} + \frac{5}{12} m - \frac{7}{20}\).

Step 1: Understand the Expression

The given expression is a sum of two algebraic expressions with fractions: \[\left(-\frac{4}{7} n^{2}+\frac{5}{6} m-\frac{1}{20}\right)+\left(\frac{3}{7} n^{2}-\frac{5}{12} m-\frac{3}{10}\right)\]Our goal is to simplify this expression.

Step 2: Combine Like Terms

First, identify and combine the like terms from both expressions:- Combine terms involving \(n^2\):\[-\frac{4}{7} n^{2} + \frac{3}{7} n^{2} = \left( -\frac{4}{7} + \frac{3}{7} \right) n^{2} = -\frac{1}{7} n^{2} \]- Combine terms involving \(m\):\[\frac{5}{6} m - \frac{5}{12} m = \left( \frac{5}{6} - \frac{5}{12} \right) m \] To simplify \(\frac{5}{6} - \frac{5}{12}\), find a common denominator, which is 12:\[\frac{10}{12} m - \frac{5}{12} m = \frac{5}{12} m\]- Combine constant terms: \[-\frac{1}{20} - \frac{3}{10} = \left( -\frac{1}{20} - \frac{6}{20} \right) = -\frac{7}{20} \]

Step 3: Present the Simplified Expression

After combining all like terms, the simplified expression becomes:\[-\frac{1}{7} n^{2} + \frac{5}{12} m - \frac{7}{20} \]This is the most simplified form of the original expression.

Key Concepts

Combining Like Terms

In algebra, similar, or "like," terms are those that have the same variable raised to the same power. When you combine like terms, you essentially add or subtract their coefficients, making the expression more simplified and manageable. In the expression provided, we first needed to locate and group the terms involving the same variables.

• For the terms involving \(n^2\), we had \(-\frac{4}{7} n^{2}\) and \(\frac{3}{7} n^{2}\). Both terms have the same variable \(n^2\), so they can be combined as \(\left( -\frac{4}{7} + \frac{3}{7} \right) n^2 = -\frac{1}{7} n^2\).
• The terms involving \(m\) were \(\frac{5}{6} m\) and \(-\frac{5}{12} m\). Again, because they share the variable \(m\), we combined them. To do this effectively, we found a common denominator and then performed the subtraction: \(\frac{10}{12} m - \frac{5}{12} m = \frac{5}{12} m\).
• Lastly, we had the constant terms \(-\frac{1}{20}\) and \(-\frac{3}{10}\). These don't have variables, but can be combined like numbers: \(-\frac{1}{20} - \frac{6}{20} = -\frac{7}{20}\).

Combining like terms is essential, as it reduces expressions to their simplest form, making them easier to understand and solve.

Fraction Operations

Fraction operations are critical when simplifying algebraic expressions, as they often involve adding or subtracting fractions. To perform these operations, you need to ensure fractions have a common denominator. This makes it possible to directly add or subtract the numerators.First, let's consider adding fractions, such as \(-\frac{4}{7} n^{2} + \frac{3}{7} n^{2}\). Since they share the same denominator (7), you only need to add the numerators: \(-4 + 3\), resulting in \(-1/7 n^2\).For fractions with different denominators, such as \(\frac{5}{6} m\) and \(-\frac{5}{12} m\), find the lowest common denominator to align them. In this case, it's 12. Convert \(\frac{5}{6} m\) to \(\frac{10}{12} m\), allowing you to subtract: \(\frac{10}{12} m - \frac{5}{12} m = \frac{5}{12} m\).It's important to always work with fractions carefully to avoid errors in calculation, particularly when handling more complex expressions or multiple fraction operations.

Algebraic Expressions

Algebraic expressions are mathematical statements that can include numbers, variables, and operations. They are the backbone of algebra and allow you to express complex mathematical ideas in a compact form.An algebraic expression can contain:

• Constants: These are fixed numbers, such as \(-\frac{1}{20}\).
• Variables: Symbols that represent unknown values, such as \(n\) and \(m\) in our expression.
• Coefficients: Numbers placed before variables, indicating how much of the variable is involved—e.g., \(-\frac{4}{7}\) is the coefficient of \(n^2\).
• Operations: Addition, subtraction, multiplication, and division, which connect the numbers and variables.

The goal in working with any algebraic expression is to simplify it by reducing it to a form where all like terms are combined and the expression is as succinct as possible. Simplification often involves fraction operations and combining like terms, as seen in the provided expression.