I need help with my Algebra Homework

Question

1.  1/2d+3/8= -2d

2.  5/2t-t= 3+3/2t

3.  -4(2-3x)= 7-2(x-3)

4.  8-2n= n+17

Answer 1

Hi Lauren:

You've got a few kinds of multistep algebra problems to do. There is a specific order that you should do the work in to make things the easiest. Here are the steps to solving multistep problems.

Print this list out and tape it to your math notebook :-)

1) Can you get rid of fractions? If you have fractions outside of parentheses or you don't have parentheses, get rid of the fractions first by multiplying by the least common denominator. 

2) Is there distribution? If there is, you must do the distribution. If there were fractions INSIDE the parentheses and they do not vanish when you do the distribution, go back to the first step and get rid of them before you continue. 

3) Can you combine like terms on each side? simplify each side before you start using addition/subtraction or multiplication/division to solve. 

4) Add or subtract as many times as you need to, to get all the constants on one side and all the variables on the other. Combine all the like terms as you go.

5) Multiply or divide to get rid of any coefficient of the variable. 

Ok, so for your specific problems.

1. 1/2d+3/8= -2d

This problem has fractions. Between 1/2 and 3/8, the common denominator is 8. Multiply every term by 8 to get rid of the fractions. 

8(1/2)d + 8(3/8) = 8(-2)d 

4d + 3 = -16d

Subtract 4d from both sides to get the variables all on one side:

3 = -20d

Divide by the coefficient, -20

-3/20 = d ... this looks weird but it will work when you sub it back in.

2. 5/2t-t= 3+3/2t

Multiply through by 2 to get rid of fractions. Remember to multiply ALL FOUR TERMS by 2, including the 3.

5t-2t=6+3t 

Combine like terms:  3t = 6 + 3t

Subtract 3t from both sides: 0 = 6 

This problem has no solution because the variable vanished and the resulting statement was not true. 

3. -4(2-3x)= 7-2(x-3)

Do exactly what Robert says in his answer. 

4. 8-2n= n+17

Whew. This one looks easier. 

First, decide what side you want the variables on. I pick the right side. So, add 2n to both sides:

8 -2n +2n = n + 17 + 2n;       8 = 3n + 17

Now subtract the 17 from both sides:

8 - 17 = 3n + 17 - 17;       -9 = 3n

Now get rid of the coefficient. 3n means 3 TIMES n. The operation that "undoes" multiplication is division, so divide by 3 to get rid of the 3. 

-9/3 = (3n)/3

-3 = n

Plug it in to check: 

8-2(-3) ?= -3+17

8+6 ?= 14

14 = 14 YAY!!!

Answer 2

I think problem 3 will let me show you a general technique that can be used for all of these problems.

-4(2-3x)= 7-2(x-3)

Let's expand everything that is in parentheses using the distributive property. On the left side of the equation, the -4 is being multiplied by everything inside those parentheses. This is how you expand them:

(-4)(2)-(-4)(3x)= 7-2(x-3)

Notice that the -4 shows up twice, once for each term inside the parentheses. I chose to place the negative sign that was in front of the 3 between the two terms. But there is another way to do it that works equally well:

(-4)(2)+(-4)(-3x)= 7-2(x-3)

Notice that the - sign stayed with the 3 in this case. But I had to put a + sign between the terms. Either way you do it, the answer is the same. Do the multiplcation and you get:

-8 + 12x = 7-2(x-3)

Remember that multiplying two negative numbers together gives you a positive number. Now lets expand the right side of the equation by distributing the 2.

-8 + 12x = 7-2(x)-2(-3)

Again, there are other ways to write this:

-8 + 12x = 7+(-2)(x)+(-2)(-3)

Using either one is okay. What is important is that both of the two's that get distributed are negative. Doing the multiplication gives us.

-8 + 12x = 7+-2x+6

Combine all the terms on each side that are similar, in this case only the 7 and the 6:

-8 + 12x = 13-2x

Now decide which side you want the x to be on and move everything else to the other side. I pick the left side for x, though I could just as easily choose the right. My first step is to get rid of the -2x on the right said. I do this by adding 2x to both sides.

-8 + 12x +2x = 13 -2x +2x

On the left, the 12x and the 2x combine to give 14x. On the right, the -2x and the 2x combine to make 0 ("cancel out").

-8 +14x = 13

Now lets get rid of the -8 on the left because we only want x on the left in the end.
 Add 8 to both sides:

-8 + 8 + 14x = 13 +8

Combine the like terms on each side. 8 and -8 give us 0 and 13 and 8 give us 21:

14x = 21

How do we get rid of the 14? Well, it is being multiplied by the x, so to get rid of it we divide both sides by 14:

14x / 14 = 21 / 14

The 14's cancel (14/14 = 1) and we are left with:

x = 21 / 14

or x = 3/2

The other problems can be solved the same way, but #1 and #2 might be easier if you start by multiplying both sides of the equation by the lowest common multiple of the denominators. That way the fractions will be gone. It is not required that you start that way, though.

Good luck.