Problem 82 - Elementary Algebra

Question

\(m\left(\frac{3}{4} m-2\right)+7\left(\frac{1}{3} m-5\right)\)

Step-by-Step Solution

Answer

\frac{3}{4} m^2 + \frac{1}{3} m - 35

Step 1 - Distribute the terms

First, distribute the terms inside the parentheses. Multiply each term inside the parentheses by the coefficients outside. The expression becomes:\[m \times \frac{3}{4} m - m \times 2 + 7 \times \frac{1}{3} m - 7 \times 5\]This simplifies to:\[\frac{3}{4} m^2 - 2m + \frac{7}{3} m - 35\]

Step 2 - Combine like terms

Next, combine the like terms. The terms involving \(m\) are \(-2m\) and \(\frac{7}{3} m\). Convert \(-2m\) to a fraction with the same denominator as \(\frac{7}{3} m\): \[-2m = -\frac{6}{3} m\]So now the expression is:\[\frac{3}{4} m^2 - \frac{6}{3} m + \frac{7}{3} m - 35\].Combining the terms with \(m\):\[\frac{3}{4} m^2 + \frac{1}{3} m - 35\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra. It allows you to multiply a term outside parentheses by each term inside the parentheses. This property is essential for simplifying expressions.
In our exercise, we start with the expression: \[m \times \frac{3}{4} m - m \times 2 + 7 \times \frac{1}{3} m - 7 \times 5\]
We apply the distributive property:

• Multiplying \[m \times \frac{3}{4} m = \frac{3}{4} m^2\]
• Multiplying\[m \times -2 = -2m\]
• Multiplying \[7 \times \frac{1}{3} m = \frac{7}{3} m\]
• Multiplying \[7 \times -5 = -35\]

Combining these distributive results, the expression simplifies to: \[\frac{3}{4} m^2 - 2m + \frac{7}{3} m - 35 \] Understanding the distributive property is key to solving more complex algebraic expressions.

Combining Like Terms

Combining like terms involves simplifying an algebraic expression by merging terms that have the same variable raised to the same power. This step helps in reducing the complexity of an expression.
In our problem, we start with: \[\frac{3}{4} m^2 - 2m + \frac{7}{3} m - 35\]
The like terms here are \(-2m\) and \(\frac{7}{3} m\). To combine them, we first need a common denominator:
\(-2m\) converts to \(-\frac{6}{3} m\):
\[\frac{3}{4} m^2 - \frac{6}{3} m + \frac{7}{3} m - 35 \]
Now, the \(m\) terms can be combined:
\[\frac{3}{4} m^2 + \frac{1}{3} m - 35\] This final step helps in achieving the simplest form of the expression, making it easier to handle in further calculations.

Fractions in Algebra

Fractions often appear in algebraic expressions and require careful handling to simplify the expressions correctly. In our exercise, we encounter fractions like \(\frac{3}{4}m^2\) and \(\frac{7}{3}m\). Understanding how to work with fractions involves:

• Converting whole numbers to fractions when needed: For instance, converting \(-2m\) to \(-\frac{6}{3}m\)
• Finding a common denominator to combine terms: The denominator of \(\frac{7}{3}m\) is 3, thus we also convert \(-2m\) to \(-\frac{6}{3}m\)
• Simplifying the fractions: Combine \(\frac{-6}{3}m\) and \(\frac{7}{3}m\) to get \(\frac{1}{3}m\)

Using these strategies helps in simplifying fractions within algebraic expressions and makes it easier to combine like terms.