Simplify The Expression: { -9(5 - 3z) =$}$ { \square$}$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. It involves combining like terms, removing unnecessary parentheses, and rearranging the expression to make it easier to work with. In this article, we will simplify the expression βˆ’9(5βˆ’3z)=β–‘{-9(5 - 3z) = \square} step by step.

Understanding the Expression

The given expression is βˆ’9(5βˆ’3z)=β–‘{-9(5 - 3z) = \square}. This expression involves a negative sign, parentheses, and variables. To simplify it, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate the expression inside the parentheses.
  2. Exponents: None in this case.
  3. Multiplication and Division: Evaluate the multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate the addition and subtraction operations from left to right.

Step 1: Distribute the Negative Sign

The first step is to distribute the negative sign to the terms inside the parentheses. When we distribute a negative sign, we change the sign of each term inside the parentheses.

βˆ’9(5βˆ’3z)=βˆ’9Γ—5+(βˆ’9)Γ—(βˆ’3z){-9(5 - 3z) = -9 \times 5 + (-9) \times (-3z)}

Step 2: Simplify the Multiplication

Now, we simplify the multiplication operations.

βˆ’9Γ—5=βˆ’45{-9 \times 5 = -45} (βˆ’9)Γ—(βˆ’3z)=27z{(-9) \times (-3z) = 27z}

Step 3: Combine Like Terms

The expression now becomes βˆ’45+27z{-45 + 27z}. We can combine the like terms by adding or subtracting the coefficients of the variables.

βˆ’45+27z=βˆ’45+27z{-45 + 27z = \boxed{-45 + 27z}}

Conclusion

In this article, we simplified the expression βˆ’9(5βˆ’3z)=β–‘{-9(5 - 3z) = \square} step by step. We followed the order of operations, distributed the negative sign, simplified the multiplication, and combined like terms. The final simplified expression is βˆ’45+27z{-45 + 27z}.

Tips and Tricks

  • When simplifying expressions, always follow the order of operations (PEMDAS).
  • Distribute the negative sign to the terms inside the parentheses.
  • Simplify the multiplication operations.
  • Combine like terms by adding or subtracting the coefficients of the variables.

Real-World Applications

Simplifying expressions is a crucial skill in mathematics that has many real-world applications. For example, in physics, we use simplifying expressions to solve problems involving motion, energy, and momentum. In engineering, we use simplifying expressions to design and optimize systems. In finance, we use simplifying expressions to calculate interest rates and investment returns.

Common Mistakes

  • Not following the order of operations (PEMDAS).
  • Not distributing the negative sign to the terms inside the parentheses.
  • Not simplifying the multiplication operations.
  • Not combining like terms.

Practice Problems

  1. Simplify the expression 2(3xβˆ’2y)=β–‘{2(3x - 2y) = \square}.
  2. Simplify the expression βˆ’4(2x+5y)=β–‘{-4(2x + 5y) = \square}.
  3. Simplify the expression 3(2xβˆ’4y)=β–‘{3(2x - 4y) = \square}.

Answer Key

  1. 6xβˆ’4y{6x - 4y}
  2. βˆ’8xβˆ’20y{-8x - 20y}
  3. 6xβˆ’12y{6x - 12y}

Conclusion

Introduction

In our previous article, we simplified the expression βˆ’9(5βˆ’3z)=β–‘{-9(5 - 3z) = \square} step by step. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS stands for:

  1. Parentheses: Evaluate the expressions inside the parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I distribute a negative sign to the terms inside the parentheses?

A: When we distribute a negative sign to the terms inside the parentheses, we change the sign of each term. For example, if we have the expression βˆ’(a+b){-(a + b)}, we can distribute the negative sign as follows:

βˆ’(a+b)=βˆ’aβˆ’b{-(a + b) = -a - b}

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change. For example, x, y, and z are variables. A constant, on the other hand, is a value that does not change. For example, 2, 5, and 10 are constants.

Q: How do I simplify an expression with multiple variables?

A: To simplify an expression with multiple variables, we need to follow the order of operations (PEMDAS) and combine like terms. For example, if we have the expression 2x+3yβˆ’4x+2y{2x + 3y - 4x + 2y}, we can simplify it as follows:

2x+3yβˆ’4x+2y=βˆ’2x+5y{2x + 3y - 4x + 2y = -2x + 5y}

Q: What is the difference between a linear expression and a quadratic expression?

A: A linear expression is an expression that contains only one variable and has a degree of 1. For example, 2x+3{2x + 3} is a linear expression. A quadratic expression, on the other hand, is an expression that contains only one variable and has a degree of 2. For example, x2+2x+1{x^2 + 2x + 1} is a quadratic expression.

Q: How do I simplify an expression with a fraction?

A: To simplify an expression with a fraction, we need to follow the order of operations (PEMDAS) and combine like terms. For example, if we have the expression 12x+13x{\frac{1}{2}x + \frac{1}{3}x}, we can simplify it as follows:

12x+13x=36x+26x=56x{\frac{1}{2}x + \frac{1}{3}x = \frac{3}{6}x + \frac{2}{6}x = \frac{5}{6}x}

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to simplifying expressions. We discussed the order of operations (PEMDAS), distributing a negative sign, variables and constants, simplifying expressions with multiple variables, linear and quadratic expressions, and simplifying expressions with fractions. We hope this article helps you improve your skills in simplifying expressions.

Practice Problems

  1. Simplify the expression 2x+3yβˆ’4x+2y{2x + 3y - 4x + 2y}.
  2. Simplify the expression 12x+13x{\frac{1}{2}x + \frac{1}{3}x}.
  3. Simplify the expression βˆ’(a+b){-(a + b)}.

Answer Key

  1. βˆ’2x+5y{-2x + 5y}
  2. 56x{\frac{5}{6}x}
  3. βˆ’aβˆ’b{-a - b}

Tips and Tricks

  • Always follow the order of operations (PEMDAS).
  • Distribute the negative sign to the terms inside the parentheses.
  • Simplify the expression by combining like terms.
  • Use variables and constants to represent values that can change and values that do not change.
  • Simplify expressions with fractions by following the order of operations (PEMDAS).