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# What are the steps that need to be taken for this equation? 5(2x-1/2)+5/2+(x+1/2)+x+3x = 15.5

What are the steps that need to be taken for this equation? 5(2x-1/2)+5/2+(x+1/2)+x+3x = 15.5.

Here is a short cut:

1. Collect all variable terms in the left, and the constant terms in the right: 15x = 15

2. Sove for x: x = 1

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I practiced this way with a 7th grader girl, and she enjoyed the short cut. You can do it too.

I would first take care of everything in the parentheses.  Expand the first term: multiply the 5 by the 2x and then by the -1/2.  The second set of parentheses can just be removed: there is nothing that it has to be multiplied by.

Then I would group all the terms with x together and add/subtract them.

Then move all numerical terms to the other side and add/subtract them as necessary.

Then divide both sides by the numerical cooefficient of the x term.

5(2x - 1/2) + 5/2 + (x + 1/2) + x + 3x = 15.5         Given

10x - 5/2 + 5/2 + x + 1/2 + x + 3x = 15.5              Distributive Property

10x + x + x + 3x - 5/2 + 5/2 + 1/2 = 15.5              Commutative Property

10x + x + x + 3x + 0 + 1/2 = 15.5                         Inverse Property of Addition

10x + x + x + 3x + 1/2 = 15.5                               Identity Property of Addition

15x + 1/2 = 15.5                                                  Simplify (10 + 1 + 1 + 3 = 15)

15x + 1/2 - 1/2 = 15.5 - 1/2                                  Subtraction property of equality

15x + 0 = 15.5 - 1/2                                             Inverse Property of Addition

15x = 15.5 - 1/2                                                   Identity Property of Addition

15x = 15                                                              Simplification (1/2 = 0.5, 15.5 - 0.5 = 15)

15x/15 = 15/15                                                     Division Property of Equality

1 * x = 1                                                              Inverse Property of Multiplication

x = 1                                                                    Identity Property of Multiplication

For this we need to follow order of operations.

Let's start with the parentheses. It looks like the terms inside can't be added together because one has an x variable attached to it while the other is just a number. The second parenthese doesn't have anything attached to it on the outside so we may remove them without altering anything. Since we can't do anything inside of the first parenthese we proceed to the next operation. In this case it's multiplication, 5(2x-1/2). The 5 will be distributed into the parenthese by multiplying both 2x and -1/2. Our result is now 10x-5/2. (Remember that when multiplying a whole number with a fraction, the whole number multiplies just the numerator).

Our resulting equation is now:

10x-5/2+5/2+x+1/2+x+3x=15.5

On the left side we can combine like terms such as 10x, x, x and 3x. 10x+x+x+3x=15x. The rest of the terms are -5/2, +5/2 and 1/2. The -5/2 and +5/2 cancel out with one another and all we have left is 1/2. Putting this together with the 15x we get:

15x+1/2=15.5

All that's left to do is to solve for x. Subtract 1/2 from each side to get 15x=15. Last, we divide each side by 15 to get x=1.

Hope this helped!

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