Simplify Each Expression:(a) $3q - 6(q - 9) + 7q = \, \square$(b) $\frac{7}{10}p - \frac{1}{12} - \frac{1}{4}p + \frac{2}{7} = \, \square$

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Introduction

In algebra, simplifying expressions is a crucial step in solving equations and manipulating mathematical statements. It involves combining like terms, removing unnecessary components, and rearranging the expression to make it more manageable. In this article, we will simplify two given expressions, (a) 3q−6(q−9)+7q= □3q - 6(q - 9) + 7q = \, \square and (b) 710p−112−14p+27= □\frac{7}{10}p - \frac{1}{12} - \frac{1}{4}p + \frac{2}{7} = \, \square, using algebraic manipulation and equation balancing techniques.

Simplifying Expression (a)

Step 1: Distribute the Negative Sign

The first step in simplifying expression (a) is to distribute the negative sign to the terms inside the parentheses.

3q−6(q−9)+7q=3q−6q+54+7q3q - 6(q - 9) + 7q = 3q - 6q + 54 + 7q

Step 2: Combine Like Terms

Next, we combine like terms, which are terms that have the same variable and exponent.

3q−6q+7q=4q3q - 6q + 7q = 4q

Step 3: Simplify the Expression

Now, we simplify the expression by combining the constant terms.

4q+544q + 54

Step 4: Final Simplified Expression

The final simplified expression is:

4q+544q + 54

Simplifying Expression (b)

Step 1: Find a Common Denominator

To simplify expression (b), we need to find a common denominator for the fractions.

The least common multiple (LCM) of 10, 12, and 4 is 60.

Step 2: Rewrite the Fractions with the Common Denominator

We rewrite each fraction with the common denominator of 60.

710p=4260p\frac{7}{10}p = \frac{42}{60}p

112=560\frac{1}{12} = \frac{5}{60}

14p=1560p\frac{1}{4}p = \frac{15}{60}p

27=1230=2460\frac{2}{7} = \frac{12}{30} = \frac{24}{60}

Step 3: Combine the Fractions

Now, we combine the fractions by adding or subtracting their numerators.

4260p−560−1560p+2460=42p−5−15p+2460\frac{42}{60}p - \frac{5}{60} - \frac{15}{60}p + \frac{24}{60} = \frac{42p - 5 - 15p + 24}{60}

Step 4: Simplify the Expression

We simplify the expression by combining like terms.

42p−15p−5+2460=27p+1960\frac{42p - 15p - 5 + 24}{60} = \frac{27p + 19}{60}

Step 5: Final Simplified Expression

The final simplified expression is:

27p+1960\frac{27p + 19}{60}

Conclusion

In conclusion, simplifying expressions is an essential skill in algebra that involves combining like terms, removing unnecessary components, and rearranging the expression to make it more manageable. By following the steps outlined in this article, we simplified two given expressions, (a) 3q−6(q−9)+7q= □3q - 6(q - 9) + 7q = \, \square and (b) 710p−112−14p+27= □\frac{7}{10}p - \frac{1}{12} - \frac{1}{4}p + \frac{2}{7} = \, \square, using algebraic manipulation and equation balancing techniques.

Tips and Tricks

  • When simplifying expressions, always look for like terms and combine them.
  • Use the distributive property to expand expressions and simplify them.
  • Find a common denominator for fractions to combine them.
  • Simplify expressions by combining like terms and removing unnecessary components.

Practice Problems

  1. Simplify the expression: 2x−5(x−3)+4x= □2x - 5(x - 3) + 4x = \, \square
  2. Simplify the expression: 34y−16−12y+23= □\frac{3}{4}y - \frac{1}{6} - \frac{1}{2}y + \frac{2}{3} = \, \square

References

Introduction

In our previous article, we simplified two given expressions, (a) 3q−6(q−9)+7q= □3q - 6(q - 9) + 7q = \, \square and (b) 710p−112−14p+27= □\frac{7}{10}p - \frac{1}{12} - \frac{1}{4}p + \frac{2}{7} = \, \square, using algebraic manipulation and equation balancing techniques. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions.

Q&A

Q: What is the first step in simplifying an expression?

A: The first step in simplifying an expression is to look for like terms and combine them. Like terms are terms that have the same variable and exponent.

Q: How do I distribute the negative sign in an expression?

A: To distribute the negative sign in an expression, you need to multiply the negative sign by each term inside the parentheses. For example, in the expression 3q−6(q−9)+7q3q - 6(q - 9) + 7q, the negative sign is distributed to the terms inside the parentheses as follows: 3q−6q+54+7q3q - 6q + 54 + 7q.

Q: What is the least common multiple (LCM) of a set of numbers?

A: The least common multiple (LCM) of a set of numbers is the smallest number that is a multiple of each of the numbers in the set. For example, the LCM of 10, 12, and 4 is 60.

Q: How do I find a common denominator for fractions?

A: To find a common denominator for fractions, you need to find the least common multiple (LCM) of the denominators. For example, to find a common denominator for the fractions 710\frac{7}{10}, 112\frac{1}{12}, and 14\frac{1}{4}, you need to find the LCM of 10, 12, and 4, which is 60.

Q: What is the difference between a like term and a unlike term?

A: A like term is a term that has the same variable and exponent, while a unlike term is a term that has a different variable or exponent. For example, in the expression 3x+4x3x + 4x, the terms 3x3x and 4x4x are like terms because they have the same variable and exponent, while the term 5y5y is a unlike term because it has a different variable.

Q: How do I simplify an expression with fractions?

A: To simplify an expression with fractions, you need to find a common denominator for the fractions and then combine them. For example, in the expression 710p−112−14p+27\frac{7}{10}p - \frac{1}{12} - \frac{1}{4}p + \frac{2}{7}, you need to find a common denominator for the fractions and then combine them.

Practice Problems

  1. Simplify the expression: 2x−5(x−3)+4x= □2x - 5(x - 3) + 4x = \, \square
  2. Simplify the expression: 34y−16−12y+23= □\frac{3}{4}y - \frac{1}{6} - \frac{1}{2}y + \frac{2}{3} = \, \square

Tips and Tricks

  • Always look for like terms and combine them when simplifying an expression.
  • Use the distributive property to expand expressions and simplify them.
  • Find a common denominator for fractions to combine them.
  • Simplify expressions by combining like terms and removing unnecessary components.

References

Conclusion

In conclusion, simplifying expressions is an essential skill in algebra that involves combining like terms, removing unnecessary components, and rearranging the expression to make it more manageable. By following the steps outlined in this article, you can simplify expressions and solve equations with ease. Remember to always look for like terms and combine them, use the distributive property to expand expressions, and find a common denominator for fractions to combine them.