Answer:
- given
- Multiplication property of equality
- Multiplicative inverse property
- Multiplicative identity property
- Addition property of equality
- Additive inverse property
- Additive identity property
Explanation:
You are given the solution to a 2-step linear equation and asked to identify the properties of equality and operations that support the steps of the solution.
Steps
5(x -3) = 20 . . . . This is the Given equation we're solving.
(1/5)(5(x -3)) = (1/5)(20) . . . . Both sides are multiplied by 1/5. The Multiplication property of equality says you can do this without changing the value of the variable.
1(x -3) = 4 . . . . (1/5)(5) is replaced by 1. The Multiplicative inverse property tells you the product of a number and its multiplicative inverse is 1.
x -3 = 4 . . . . 1(x -3) is replaced by (x -3). The Multiplicative identity property tells you multiplication by 1 changes nothing. These properties of multiplication are what allow you to remove the factor of 5 from the equation.
Now, we're about to do the same thing with addition: use its properties to remove the unwanted -3.
x -3 +3 = 4 +3 . . . . 3 is added to both sides. The Addition property of equality says you can do this without changing the value of the variable.
x +0 = 7 . . . . -3+3 is replaced by 0. The Additive inverse property tells you the sum of a number and its additive inverse is zero.
x = 7 . . . . x+0 is replaced by x. The Additive identity property tells you addition of 0 changes nothing. These properties of addition are what allow you to remove the added -3 from the equation.
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Additional comment
Clearly, for exercises like this, it is essential to know the names of these properties and what they mean. Once you understand cancelling multiplication using multiplicative inverses, and cancelling addition using additive inverses, you are on your way to solving most algebra problems.
You must always keep in mind the properties of equality that tell you the same operation must be performed on both sides of the equal sign.
Understanding these properties can also help you understand the relation between multiplication and division, addition and subtraction. Division is multiplication by the multiplicative inverse; subtraction is addition of the additive inverse.