Simplify The Following Expressions:1. $ (-0.5z) - (0.4z + 7) + (0.3z + 7) $2. $ \left(3 \frac{1}{2} Y - \frac{2}{5}\right) + \left(1 \frac{2}{5} - 2 \frac{3}{4} Y\right) - \left(5y - \frac{3}{5}\right) $

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore two complex algebraic expressions and simplify them step by step. We will use the distributive property, combine like terms, and apply other algebraic rules to simplify the expressions.

Simplifying the First Expression

The first expression is:

$(-0.5z) - (0.4z + 7) + (0.3z + 7)$

To simplify this expression, we will start by applying the distributive property to the second and third terms.

Distributive Property

The distributive property states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$a(b + c) = ab + ac$

We can apply this property to the second and third terms of the expression:

$(-0.5z) - (0.4z + 7) + (0.3z + 7)$

$= (-0.5z) - 0.4z - 7 + 0.3z + 7$

Combining Like Terms

Now that we have applied the distributive property, we can combine like terms. Like terms are terms that have the same variable raised to the same power.

In this expression, we have the following like terms:

• $-0.5z$ and $-0.4z$ are like terms because they both have the variable $z$ raised to the power of 1.
• $-7$ and $+7$ are like terms because they are both constants.

We can combine these like terms as follows:

$= (-0.5z - 0.4z) + (-7 + 7)$

$= -0.9z + 0$

Final Simplification

The final simplified expression is:

$-0.9z$

This is the simplified form of the first expression.

Simplifying the Second Expression

The second expression is:

$\left(3 \frac{1}{2} y - \frac{2}{5}\right) + \left(1 \frac{2}{5} - 2 \frac{3}{4} y\right) - \left(5y - \frac{3}{5}\right)$

To simplify this expression, we will start by converting the mixed numbers to improper fractions.

Converting Mixed Numbers to Improper Fractions

A mixed number is a combination of a whole number and a fraction. We can convert a mixed number to an improper fraction by multiplying the whole number by the denominator and adding the numerator.

In this expression, we have the following mixed numbers:

• $3 \frac{1}{2} y = \frac{7}{2} y$
• $1 \frac{2}{5} = \frac{7}{5}$
• $2 \frac{3}{4} y = \frac{11}{4} y$

We can convert these mixed numbers to improper fractions as follows:

$\left(\frac{7}{2} y - \frac{2}{5}\right) + \left(\frac{7}{5} - \frac{11}{4} y\right) - \left(5y - \frac{3}{5}\right)$

Distributive Property

We can apply the distributive property to the second and third terms of the expression:

$\left(\frac{7}{2} y - \frac{2}{5}\right) + \left(\frac{7}{5} - \frac{11}{4} y\right) - \left(5y - \frac{3}{5}\right)$

$= \frac{7}{2} y - \frac{2}{5} + \frac{7}{5} - \frac{11}{4} y - 5y + \frac{3}{5}$

Combining Like Terms

Now that we have applied the distributive property, we can combine like terms. Like terms are terms that have the same variable raised to the same power.

In this expression, we have the following like terms:

• $\frac{7}{2} y$ and $-\frac{11}{4} y$ and $-5y$ are like terms because they all have the variable $y$ raised to the power of 1.
• $-\frac{2}{5}$ and $\frac{7}{5}$ and $\frac{3}{5}$ are like terms because they are all constants.

We can combine these like terms as follows:

$= \left(\frac{7}{2} y - \frac{11}{4} y - 5y\right) + \left(-\frac{2}{5} + \frac{7}{5} + \frac{3}{5}\right)$

$= \left(\frac{14}{4} y - \frac{11}{4} y - \frac{20}{4} y\right) + \left(\frac{-2 + 7 + 3}{5}\right)$

$= \left(\frac{3}{4} y\right) + \left(\frac{8}{5}\right)$

Final Simplification

The final simplified expression is:

$\frac{3}{4} y + \frac{8}{5}$

This is the simplified form of the second expression.

Conclusion

Introduction

In our previous article, we explored two complex algebraic expressions and simplified them step by step. We applied the distributive property, combined like terms, and used other algebraic rules to simplify the expressions. In this article, we will answer some common questions related to simplifying algebraic expressions.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand a single term into multiple terms. It states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$a(b + c) = ab + ac$

Q: How do I apply the distributive property?

A: To apply the distributive property, simply multiply the single term by each of the terms inside the parentheses. For example, if we have the expression $2(x + 3)$, we can apply the distributive property as follows:

$2(x + 3) = 2x + 6$

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, in the expression $2x + 3x$, the terms $2x$ and $3x$ are like terms because they both have the variable $x$ raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, simply add or subtract the coefficients of the like terms. For example, if we have the expression $2x + 3x$, we can combine the like terms as follows:

$2x + 3x = 5x$

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when simplifying an expression. The order of operations is as follows:

1. Parentheses: Evaluate expressions inside 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 simplify a complex expression?

A: To simplify a complex expression, follow these steps:

1. Apply the distributive property to expand any single terms.
2. Combine like terms.
3. Simplify any exponential expressions.
4. Evaluate any multiplication and division operations from left to right.
5. Finally, evaluate any addition and subtraction operations from left to right.

Q: What are some common algebraic rules?

A: Some common algebraic rules include:

• The distributive property: $a(b + c) = ab + ac$
• The commutative property: $a + b = b + a$
• The associative property: $(a + b) + c = a + (b + c)$
• The identity property: $a + 0 = a$
• The inverse property: $a + (-a) = 0$

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By understanding the distributive property, combining like terms, and applying other algebraic rules, we can simplify even the most complex expressions. In this article, we have answered some common questions related to simplifying algebraic expressions and provided a step-by-step guide to simplifying complex expressions.