# Solve the following algebraic expression:[5z-(x+2y)]-[3x-(y-2z)]

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Greetings. This algebraic expression (incidentally, what's the difference between an expression and an equation? Answer follows) requires you to see that a negative sign can be interpreted as (-1) and distributed using the Distributive Property of Multiplication.

Using the problem-solving strategy of "Simplify and Solve," we first look to determine if we can combine similar terms. Not yet. So, let's address those nasty (-) minus signs. To eliminate the confusion that they create, shall we change all operation signs to (+) addition signs? Let's do that:

[5z + -1(x + 2y)] + -1[3x + -1(y + -2z)]

= 5z + -1x + -2y + -1[3x + -1y + 2z]

= 5z + -1x + -2y + -3x + 1y + -2z Now we combine similar terms

= -4x + -y + 3z

(Oh yes! An expression does not include an equal sign, but an equation does include an equal sign).

This problem is about moving the minus sign inside the parenthesis so that order of operations is preserved.

[5z-(x+2y)]-[3x-(y-2z)]

5z - x - 2y - (3x - y + 2z)

5z - x - 2y - 3x + y - 2z

3z - 4x - y

Note:

Many algebra textbooks suggest that variables should written in alphabetical order with decreasing exponents with secondary variables having increasing exponents. For example,

x^2 +2xy + y^2 is proper; however

y^2 + x^2 +2xy is not.

For the original problem described above, a better answer would be

- 4x - y + 3z

instead of demanding that the first term always be positive.

[5z-(x+2y)]-[3x-(y-2z)].

Hope this expression is to be simplified.

The expression in first bracket,[5z-(x+2y)] = 5z-x-2y, as -x(+2y) = -1*x + (-1)(2y) = -x-2y.

The expression in the 2nd bracket,[3x-(y-2z)] = [3x-y--2z]=[3x-y+2z]

So the given expression is now equal to

5z-x-2y - [3x-y+2z]

=5z-x-2y-3x--y-2z.

5z-x-2y-3x+y-2z.Collecting the like terms, we get,

=-x-3x - 2y+y + 5z-2z

=-4x-y+3z or 3z-4x-y

This expression is testing your knowledge of the order of operations and the concept of like terms. The order of operations states that you should work from the inside out when removing grouping symbols. Basically, if a negative occurs in front of a grouping symbol, meaning (), {}, [], etc., it is telling you to take the negative sign into the grouping symbol. This process changes the sign of everything inside the grouping symbols.

Additionally, you have to remember the basic rules of multiplying negatives and positives. If you are multiplying a negative and a negative you answer will be positive. Likewise, negative times a positive will give you a negative answer. Basically the rules mean: If you have an odd number of negative signs in your multiplication problem you will have a negative answer and an even number of negative signs will yield a positive answer. Your expression would be simplified as follows:

[5z – (x + 2y)] – [3x –(y – 2z)] be careful with your signs

[5z – x – 2y] – [3x – y + 2z] drop the first grouping symbol

and apply the negative the

second

5z – x – 2y – 3x + y – 2z rearrange your like terms and

combine

5z – 2z – x – 3x – 2y + y

3z - 4x - y