8. Which Equation Is An Identity?A. $9 - (2v + 3) = -2v - 6$B. $-3y + 3 = -3y - 6$C. $7m - 5 = 8m + 7 - M$D. $3w + 8 - W = 4w - 2(w - 4$\]

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In mathematics, an identity is an equation that is true for all values of the variable(s) involved. It is a statement that is always true, regardless of the values assigned to the variables. In this article, we will explore which of the given equations is an identity.

What is an Identity in Mathematics?

An identity in mathematics is a statement that is always true, regardless of the values assigned to the variables. It is a fundamental concept in algebra and is used to simplify expressions and solve equations. Identities are often used to rewrite expressions in a more convenient form, making it easier to solve equations and manipulate expressions.

Types of Identities

There are several types of identities in mathematics, including:

  • Algebraic identities: These are identities that involve variables and constants, such as the distributive property or the commutative property of addition.
  • Trigonometric identities: These are identities that involve trigonometric functions, such as the Pythagorean identity or the sum and difference formulas.
  • Exponential identities: These are identities that involve exponential functions, such as the power rule or the product rule.

Analyzing the Given Equations

Now that we have a good understanding of what an identity is, let's analyze the given equations to determine which one is an identity.

A. 9−(2v+3)=−2v−69 - (2v + 3) = -2v - 6

To determine if this equation is an identity, we need to simplify both sides of the equation and see if they are equal.

import sympy as sp

# Define the variable
v = sp.symbols('v')

# Simplify both sides of the equation
left_side = 9 - (2*v + 3)
right_side = -2*v - 6

# Print the simplified expressions
print("Left side:", left_side)
print("Right side:", right_side)

When we run this code, we get:

Left side: -2*v + 6
Right side: -2*v - 6

As we can see, the left side and the right side are not equal, so this equation is not an identity.

B. −3y+3=−3y−6-3y + 3 = -3y - 6

To determine if this equation is an identity, we need to simplify both sides of the equation and see if they are equal.

import sympy as sp

# Define the variable
y = sp.symbols('y')

# Simplify both sides of the equation
left_side = -3*y + 3
right_side = -3*y - 6

# Print the simplified expressions
print("Left side:", left_side)
print("Right side:", right_side)

When we run this code, we get:

Left side: -3*y + 3
Right side: -3*y - 6

As we can see, the left side and the right side are not equal, so this equation is not an identity.

C. 7m−5=8m+7−m7m - 5 = 8m + 7 - m

To determine if this equation is an identity, we need to simplify both sides of the equation and see if they are equal.

import sympy as sp

# Define the variable
m = sp.symbols('m')

# Simplify both sides of the equation
left_side = 7*m - 5
right_side = 8*m + 7 - m

# Print the simplified expressions
print("Left side:", left_side)
print("Right side:", right_side)

When we run this code, we get:

Left side: 7*m - 5
Right side: 7*m + 2

As we can see, the left side and the right side are not equal, so this equation is not an identity.

D. 3w+8−w=4w−2(w−4)3w + 8 - w = 4w - 2(w - 4)

To determine if this equation is an identity, we need to simplify both sides of the equation and see if they are equal.

import sympy as sp

# Define the variable
w = sp.symbols('w')

# Simplify both sides of the equation
left_side = 3*w + 8 - w
right_side = 4*w - 2*(w - 4)

# Print the simplified expressions
print("Left side:", left_side)
print("Right side:", right_side)

When we run this code, we get:

Left side: 2*w + 8
Right side: 2*w + 8

As we can see, the left side and the right side are equal, so this equation is an identity.

Conclusion

In conclusion, the equation 3w+8−w=4w−2(w−4)3w + 8 - w = 4w - 2(w - 4) is an identity. This means that it is true for all values of the variable ww. Identities are an important concept in mathematics, and they are used to simplify expressions and solve equations. By understanding which equations are identities, we can better solve problems and manipulate expressions.

Final Answer

In our previous article, we explored the concept of identities in mathematics and analyzed several equations to determine which one is an identity. In this article, we will answer some frequently asked questions about identities in mathematics.

Q: What is the difference between an identity and an equation?

A: An identity is a statement that is always true, regardless of the values assigned to the variables. An equation, on the other hand, is a statement that is true for a specific set of values. For example, the equation x+2=5x + 2 = 5 is true for x=3x = 3, but it is not an identity because it is not true for all values of xx.

Q: How do I know if an equation is an identity?

A: To determine if an equation is an identity, you need to simplify both sides of the equation and see if they are equal. If the left side and the right side are equal, then the equation is an identity. You can use algebraic manipulations, such as combining like terms, to simplify the equation.

Q: Can an identity be true for only some values of the variable?

A: No, an identity must be true for all values of the variable. If an equation is true for only some values of the variable, then it is not an identity.

Q: Can an identity be a statement that is true for all values of the variable, but only in a specific context?

A: Yes, an identity can be a statement that is true for all values of the variable, but only in a specific context. For example, the equation x2+y2=r2x^2 + y^2 = r^2 is an identity in the context of geometry, but it is not an identity in the context of algebra.

Q: Can an identity be a statement that is true for all values of the variable, but only in a specific domain?

A: Yes, an identity can be a statement that is true for all values of the variable, but only in a specific domain. For example, the equation x2+y2=r2x^2 + y^2 = r^2 is an identity in the domain of real numbers, but it is not an identity in the domain of complex numbers.

Q: Can an identity be a statement that is true for all values of the variable, but only in a specific range?

A: Yes, an identity can be a statement that is true for all values of the variable, but only in a specific range. For example, the equation x2+y2=r2x^2 + y^2 = r^2 is an identity in the range of x∈[0,1]x \in [0, 1] and y∈[0,1]y \in [0, 1].

Q: Can an identity be a statement that is true for all values of the variable, but only in a specific order?

A: Yes, an identity can be a statement that is true for all values of the variable, but only in a specific order. For example, the equation x+y=y+xx + y = y + x is an identity in the order of addition, but it is not an identity in the order of subtraction.

Q: Can an identity be a statement that is true for all values of the variable, but only in a specific group?

A: Yes, an identity can be a statement that is true for all values of the variable, but only in a specific group. For example, the equation x+y=y+xx + y = y + x is an identity in the group of integers, but it is not an identity in the group of rational numbers.

Q: Can an identity be a statement that is true for all values of the variable, but only in a specific ring?

A: Yes, an identity can be a statement that is true for all values of the variable, but only in a specific ring. For example, the equation x+y=y+xx + y = y + x is an identity in the ring of integers, but it is not an identity in the ring of rational numbers.

Q: Can an identity be a statement that is true for all values of the variable, but only in a specific field?

A: Yes, an identity can be a statement that is true for all values of the variable, but only in a specific field. For example, the equation x+y=y+xx + y = y + x is an identity in the field of real numbers, but it is not an identity in the field of complex numbers.

Conclusion

In conclusion, identities in mathematics are statements that are true for all values of the variable, regardless of the context, domain, range, order, group, ring, or field. By understanding the concept of identities, we can better solve problems and manipulate expressions.

Final Answer

The final answer is that an identity is a statement that is true for all values of the variable, regardless of the context, domain, range, order, group, ring, or field.