Question

Hard problem please help

Identify the variable.
Find out the operation in the problem.
Use the inverse operation on both sides of the equation to solve.

Answer 1

1. m; addition; m=86
2. x; addition; x=1
3. b; multiplication; b=-9
4. x; multiplication; x=-70

Explanation:

The variable is the letter. The operation is shown by the math symbol being applied to the letter.

The inverse of addition is addition of the additive inverse (opposite).

The inverse of multiplication is multiplication by the multiplicative inverse (reciprocal).

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1.

m -38 = 48

Variable: m

Operation: addition of -38

Showing use of inverse operation:

m -38 +38 = 48 +38 . . . . . . addition of the opposite of -38

m = 86

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2.

x +32 = 33

Variable: x

Operation: addition of 32

Showing use of inverse operation:

x +32 +(-32) = 33 +(-32)

x = 1

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3.

-6b = 54

Variable: b

Operation: multiplication by -6

Showing use of inverse operation:

-6b(-1/6) = 54(-1/6) . . . . . . multiplication by the reciprocal of -6

b = -9

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4.

x/10 = -7

Variable: x

Operation: multiplication by 1/10

Showing use of the inverse operation:

x/10(10) = -7(10) . . . . . . multiplication by the reciprocal of 1/10

x = -70

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Additional comment

Here, we have chosen to represent subtraction and division as addition and multiplication respectively. This is partly to simplify the explanation in the answer, and partly to drive home the notions that ...

• subtraction is addition of the opposite (additive inverse)
• division is multiplication by the reciprocal (multiplicative inverse)

The purpose of an inverse operation is to give you the identity element for that operation. When the operation is performed using its identity element, the original value is unchanged (retains its original identity).

In problem 1, m -38 +38 = m +0 = m. additive identity element: 0

In problem 4, (x/10)(10) = x(1) = x. multiplicative identity element: 1