/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 82 \(m\left(\frac{3}{4} m-2\right)+... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 82

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

Short Answer

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

Step by step solution

01

- 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:\[\frac{3}{4} m^2 - 2m + \frac{7}{3} m - 35\]
02

- 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 \(m\):\[\frac{3}{4} m^2 + \frac{1}{3} m - 35\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

A drafter is making enlargements of a rectangular drawing that preserve the relative width and length of the drawing. The length of the drawing is five- fourths of the width. a. If \(W=\) width, write a polynomial expression in \(W\) that represents the length, and draw a diagram of the rectangle. Do not include the units. b. Write a polynomial expression in \(W\) that represents the perimeter. c. Write a polynomial expression in \(W\) that represents the area.

The base of a triangle is 4 in. longer than twice the height. a. If \(h=\) height, write a polynomial expression in \(h\) that represents the base, and draw a diagram of the triangle. Do not include the units. b. Write a polynomial expression in \(h\) that represents the area.

\(\left(9 x^{2}-6 x+1\right) \div(3 x-1)\)

\(\left(40 x^{2}+122 x+55\right) \div(2 x+5)\)

\(\left(72 v^{6}-81 v^{4}+54 v^{2}\right) \div\left(9 v^{2}\right)\)
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