Question

Solve each step and tell which property is used, each is four steps

2x - 5 = 8x + 7
4.5 - 7 = 2(y + 2.25) + 6.6
-9 + 6x = -3(3 - 2x)
7(x+4) = 5(x+2)
-6y + 8 = y +9 - 7y - 1

Answer 1

Define the following
Properties of Equality, Distributive property, Identity property

Answer 2

Solve for each of the formulas for the indicated variable:
P = 2L + 2W for W

ax + b = 0 for x

V = L W H for H

y = mx + b for x

If the perimeter of a rectangle is 60 ft and its length is 18 ft, find its width. (see formula in number 1)

Answer 3

define the following
Formula, Rearrange an equation

Answer 4

Let's solve each equation step-by-step and identify the properties used.

### 1. Equation: \( 2x - 5 = 8x + 7 \)

**Step 1**: Add \( 5 \) to both sides.
\( 2x - 5 + 5 = 8x + 7 + 5 \)
\( 2x = 8x + 12 \)
**Property Used**: Addition Property of Equality

**Step 2**: Subtract \( 8x \) from both sides.
\( 2x - 8x = 8x + 12 - 8x \)
\( -6x = 12 \)
**Property Used**: Addition Property of Equality

**Step 3**: Divide both sides by \( -6 \).
\( \frac{-6x}{-6} = \frac{12}{-6} \)
\( x = -2 \)
**Property Used**: Division Property of Equality

**Step 4**: The solution is \( x = -2 \).

---

### 2. Equation: \( 4.5 - 7 = 2(y + 2.25) + 6.6 \)

**Step 1**: Simplify the left side.
\( 4.5 - 7 = -2.5 \)
**Property Used**: Simplification

**Step 2**: Distribute \( 2 \) on the right side.
\( 2(y + 2.25) = 2y + 4.5 \)
\( -2.5 = 2y + 4.5 + 6.6 \)
**Property Used**: Distributive Property

**Step 3**: Combine like terms on the right side.
\( 4.5 + 6.6 = 11.1 \)
\( -2.5 = 2y + 11.1 \)
**Property Used**: Combining Like Terms

**Step 4**: Subtract \( 11.1 \) from both sides.
\( -2.5 - 11.1 = 2y \)
\( -13.6 = 2y \)
**Property Used**: Addition Property of Equality

**Step 5**: Divide both sides by \( 2 \).
\( y = \frac{-13.6}{2} = -6.8 \)
**Property Used**: Division Property of Equality

The solution is \( y = -6.8 \).

---

### 3. Equation: \( -9 + 6x = -3(3 - 2x) \)

**Step 1**: Distribute \( -3 \) on the right side.
\( -9 + 6x = -9 + 6x \)
**Property Used**: Distributive Property

**Step 2**: Add \( 9 \) to both sides.
\( -9 + 9 + 6x = -9 + 9 + 6x \)
\( 6x = 6x \)
**Property Used**: Addition Property of Equality

**Step 3**: Subtract \( 6x \) from both sides.
\( 6x - 6x = 6x - 6x \)
\( 0 = 0 \)
**Property Used**: Subtraction Property of Equality.

**Conclusion**: This indicates that the equation is an identity, true for all \( x \).

---

### 4. Equation: \( 7(x + 4) = 5(x + 2) \)

**Step 1**: Distribute on both sides.
\( 7x + 28 = 5x + 10 \)
**Property Used**: Distributive Property

**Step 2**: Subtract \( 5x \) from both sides.
\( 7x - 5x + 28 = 10 \)
\( 2x + 28 = 10 \)
**Property Used**: Subtraction Property of Equality

**Step 3**: Subtract \( 28 \) from both sides.
\( 2x + 28 - 28 = 10 - 28 \)
\( 2x = -18 \)
**Property Used**: Addition Property of Equality

**Step 4**: Divide both sides by \( 2 \).
\( x = \frac{-18}{2} \)
\( x = -9 \)
**Property Used**: Division Property of Equality

The solution is \( x = -9 \).

---

### 5. Equation: \( -6y + 8 = y + 9 - 7y - 1 \)

**Step 1**: Simplify the right side.
Combine like terms \( y - 7y + 9 - 1 \) gives:
\( -6y + 8 = -6y + 8 \)
**Property Used**: Combining Like Terms

**Step 2**: Add \( 6y \) to both sides.
\( -6y + 6y + 8 = -6y + 6y + 8 \)
\( 8 = 8 \)
**Property Used**: Addition Property of Equality

**Conclusion**: This indicates that the equation is an identity, true for all \( y \).

---

These steps solve each equation while clearly indicating the properties used during manipulation.

Answer 5

Here are definitions for the Properties of Equality, the Distributive Property, and the Identity Property:

### 1. Properties of Equality

The **Properties of Equality** refer to a set of rules in algebra that state how equalities can be manipulated while retaining their truth. These properties ensure that if two expressions are equal, any operation applied to both sides will maintain that equality. The fundamental properties include:

- **Addition Property of Equality**: If \( a = b \), then \( a + c = b + c \) for any number \( c \). This means you can add the same value to both sides of an equation.

- **Subtraction Property of Equality**: If \( a = b \), then \( a - c = b - c \) for any number \( c \). This means you can subtract the same value from both sides of an equation.

- **Multiplication Property of Equality**: If \( a = b \), then \( a \cdot c = b \cdot c \) for any number \( c \). This means you can multiply both sides of an equation by the same value.

- **Division Property of Equality**: If \( a = b \) and \( c \neq 0 \), then \( \frac{a}{c} = \frac{b}{c} \). This means you can divide both sides of an equation by the same value as long as you're not dividing by zero.

### 2. Distributive Property

The **Distributive Property** is a property of multiplication over addition (or subtraction). It states that multiplying a number by a sum (or difference) is the same as multiplying each addend (or terms in the case of subtraction) individually by that number and then adding (or subtracting) the results:

\[
a(b + c) = ab + ac
\]
\[
a(b - c) = ab - ac
\]

In this context:
- \( a \) is the multiplier,
- \( b \) and \( c \) are the addends (or terms).

This property is essential for simplifying expressions and solving equations.

### 3. Identity Property

The **Identity Property** states that there exists certain numbers that, when combined with any number in a specific operation, result in the same original number. There are two main types:

- **Additive Identity Property**: The additive identity is \( 0 \). This means that any number plus \( 0 \) remains unchanged:

\[
a + 0 = a
\]

- **Multiplicative Identity Property**: The multiplicative identity is \( 1 \). This means that any number multiplied by \( 1 \) remains unchanged:

\[
a \cdot 1 = a
\]

These properties are crucial for maintaining the integrity of equations as they are manipulated in algebraic processes.

---

If you have any more questions or need further explanations on these properties, feel free to ask!

Answer 6

Let's solve each formula for the indicated variable step-by-step.

### 1. Solve \( P = 2L + 2W \) for \( W \)

**Step 1**: Start with the equation.
\[
P = 2L + 2W
\]

**Step 2**: Subtract \( 2L \) from both sides to isolate the term with \( W \).
\[
P - 2L = 2W
\]

**Step 3**: Divide both sides by \( 2 \) to solve for \( W \).
\[
W = \frac{P - 2L}{2}
\]

So, the formula solved for \( W \) is:
\[
W = \frac{P - 2L}{2}
\]

---

### 2. Solve \( ax + b = 0 \) for \( x \)

**Step 1**: Start with the equation.
\[
ax + b = 0
\]

**Step 2**: Subtract \( b \) from both sides.
\[
ax = -b
\]

**Step 3**: Divide both sides by \( a \) (assuming \( a \neq 0 \)).
\[
x = \frac{-b}{a}
\]

So, the formula solved for \( x \) is:
\[
x = \frac{-b}{a}
\]

---

### 3. Solve \( V = L W H \) for \( H \)

**Step 1**: Start with the equation.
\[
V = L W H
\]

**Step 2**: Divide both sides by \( L W \) (assuming \( L \neq 0 \) and \( W \neq 0 \)).
\[
H = \frac{V}{L W}
\]

So, the formula solved for \( H \) is:
\[
H = \frac{V}{L W}
\]

---

### 4. Solve \( y = mx + b \) for \( x \)

**Step 1**: Start with the equation.
\[
y = mx + b
\]

**Step 2**: Subtract \( b \) from both sides.
\[
y - b = mx
\]

**Step 3**: Divide both sides by \( m \) (assuming \( m \neq 0 \)).
\[
x = \frac{y - b}{m}
\]

So, the formula solved for \( x \) is:
\[
x = \frac{y - b}{m}
\]

---

### 5. Finding the Width of a Rectangle

Given:
- The perimeter \( P = 60 \) ft
- The length \( L = 18 \) ft

Using the formula for the perimeter of a rectangle:
\[
P = 2L + 2W
\]

**Step 1**: Substitute the known values into the formula.
\[
60 = 2(18) + 2W
\]

**Step 2**: Calculate \( 2(18) \).
\[
60 = 36 + 2W
\]

**Step 3**: Subtract \( 36 \) from both sides.
\[
60 - 36 = 2W
\]
\[
24 = 2W
\]

**Step 4**: Divide both sides by \( 2 \).
\[
W = \frac{24}{2}
\]
\[
W = 12 \text{ ft}
\]

So, the width of the rectangle is \( 12 \) ft.

Answer 7

Here are definitions for **Formula** and **Rearranging an Equation**:

### 1. Formula

A **formula** is a mathematical expression that describes a relationship between different quantities or variables. It is typically composed of symbols, constants, variables, and mathematical operations (such as addition, subtraction, multiplication, or division). Formulas are often used to calculate values or to express relationships in a compact way.

For example:
- The area \( A \) of a rectangle can be expressed as a formula:
\[
A = L \times W
\]
where \( L \) is the length and \( W \) is the width of the rectangle.

Formulas are commonly found in various fields such as physics, chemistry, finance, and engineering, serving as tools to solve problems involving quantitative relationships.

### 2. Rearranging an Equation

**Rearranging an equation** refers to the process of manipulating an equation to isolate a specific variable on one side of the equation. This involves applying the properties of equality, arithmetic operations, and algebraic techniques to transform the original equation while maintaining its equality.

The goal of rearranging is to solve for a particular variable, which can be helpful in various mathematical problems.

For example, if we have the equation \( ax + b = c \) and want to rearrange it to solve for \( x \):

1. Subtract \( b \) from both sides:
\[
ax = c - b
\]

2. Then divide both sides by \( a \) (assuming \( a \neq 0 \)):
\[
x = \frac{c - b}{a}
\]

By rearranging the equation, we have successfully expressed \( x \) in terms of \( a \), \( b \), and \( c \). Rearranging equations is a fundamental skill in algebra and helps in understanding the relationships between different variables.

If you need further information or examples, feel free to ask!