Is this equation an identity?
[tex]
\begin{array}{l}
-1+4(3 d-13)=18 d+13+ \
18
\end{array}
[/tex]
yes
no
Answer (2)
Simplify the left-hand side of the equation: − 1 + 4 ( 3 d − 13 ) = 12 d − 53 .
Simplify the right-hand side of the equation: 18 d + 13 + 18 = 18 d + 31 .
Compare the simplified expressions: 12 d − 53 and 18 d + 31 .
Since 12 d − 53 is not equal to 18 d + 31 for all values of d , the equation is not an identity. n o
Explanation
Understanding the Problem We are given the equation: − 1 + 4 ( 3 d − 13 ) = 18 d + 13 + 18 Our goal is to determine whether this equation is an identity. An identity is an equation that is true for all values of the variable d . To check if the given equation is an identity, we need to simplify both sides of the equation and see if they are equal.
Simplifying the Left-Hand Side First, let's simplify the left-hand side (LHS) of the equation: − 1 + 4 ( 3 d − 13 ) = − 1 + 4 ( 3 d ) + 4 ( − 13 ) = − 1 + 12 d − 52 = 12 d − 53 So, the simplified LHS is 12 d − 53 .
Simplifying the Right-Hand Side Next, let's simplify the right-hand side (RHS) of the equation: 18 d + 13 + 18 = 18 d + 31 So, the simplified RHS is 18 d + 31 .
Comparing LHS and RHS Now, we compare the simplified LHS and RHS: 12 d − 53 = 18 d + 31 If this equation is an identity, then the LHS and RHS must be equal for all values of d . Let's test a few values of d :
If d = 0 , then LHS = 12 ( 0 ) − 53 = − 53 and RHS = 18 ( 0 ) + 31 = 31 . Since − 53 e q 31 , the equation is not an identity. If d = 1 , then LHS = 12 ( 1 ) − 53 = − 41 and RHS = 18 ( 1 ) + 31 = 49 . Since − 41 e q 49 , the equation is not an identity. If d = 2 , then LHS = 12 ( 2 ) − 53 = 24 − 53 = − 29 and RHS = 18 ( 2 ) + 31 = 36 + 31 = 67 . Since − 29 e q 67 , the equation is not an identity. Since the LHS and RHS are not equal for all values of d , the equation is not an identity.
Conclusion Since the simplified left-hand side ( 12 d − 53 ) is not equal to the simplified right-hand side ( 18 d + 31 ) for all values of d , the given equation is not an identity. Therefore, the answer is no.
Examples
In electrical engineering, when designing circuits, it's crucial to verify if two different circuit configurations are equivalent. This involves checking if the equations representing the behavior of the circuits are identities. If the equations are identical, the circuits behave the same way, simplifying the design and analysis process. For example, simplifying complex circuit equations to check for equivalence can save time and resources by ensuring that different designs achieve the same desired outcome.
The equation is not an identity because the left-hand side, after simplification, is 12 d − 53 , which is not equal to the right-hand side 18 d + 31 for all values of d . Thus, the answer is no.
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