One-Step Equations

One-step equations are the simplest equations. They can be solved with minmal effort and in "one-step". Let's take a look:

Examples:

1. 7x = 21



To solve this equation we need to figure out what x would be to make it true. Since the 7 and the x are tied together by multiplication, the only way to undo it is to divide. In order to get x alone we would need to divide by 7. Once we do this operation to the left side we have to do the same operation to the right side. It would look like this:



7x / 7
=
21 / 7
 , solving we get x=3.



2. -10x = 20



Once again, we see that the -10 and the x are tied together by multiplication, the only way to undo it is to divide. In order to get x alone we would need to divide by -10. Once we do this operation to the left side we have to do the same operation to the right side. It would look like this:



-10x / -10
=
20 / -10
 , solving we get x=-2. Notice that are answer is negative, since we divided a positive number and negative number.



3. 3 / 4
   x = 6

   To solve this equation, we need to undo the fraction. We need to get rid of the denominator, so we are only left with the numerator. In order to get rid of the fraction we need to multiply, since it is tied to the x by division. We need to multiply by the reciprocal. Once we do that on the left side, we need to do it to the right side.
   

   Step 1: Multiply by the reciprocal.
   (
   4 / 3
   ) (
   3 / 4
   ) x = 6(
   4 / 3
   )
   We now have: x =
   24 / 3
   
   Simplifying we get the solution of x = 8.
   We can check our solution by plugging 8 back into the equation where we see the x.
   (
   3 / 4
   )* 8 = 6
   
   24 / 3
   = 6
   6 = 6 ✔

Two-Step Equations

Two-step equations are a little bit more complex. They can be solved with a bit more thinking. These problems can be solved in "two-steps". Let's take a look:

Examples:

1. 2x + 3 = 7

To solve this equation we use the same process that we used in the one step equations. We need to get x by itself. How would we do this?

• Step 1. Subtract 3 from both sides.
2x + 3 - 3 = 7 - 3
2x = 4
• Step 2. Divide by 2.
  
  2x / 2
  =
  4 / 2
• Step 3. Rewrite the equation.
  x = 2
• Step 4. Verify the solution.
  2(2) + 3 = 7
  4 + 3 = 7
  7 = 7 ✔

Now for something a little trickier.
• 3 / 2
  x - 5 = 10
  
  First we want to get x alone. We can notice that there is (-5) on the right side, the same side as the x. What do we do now? In order to get rid of the negative number we need to add it to both sides.
• Step 1: Add 5 to both sides.
3 / 2
x - 5 + 5 = 10 + 5


We are now left with



3 / 2
x = 15

• Step 2. We now need to isolate the x. We notice that it is attached by a fraction. We can get rid of the fraction one of two ways. First we can multiply by the reciprocal, similar to the last problem in the one-step section. Another way is to multiply by the reciprocal of the divisor. It would look something like this:
  (
  2 / 1
  ) (
  3 / 2
  ) x = 15(
  2 / 1
  )
  We now have: 3x =
  30 / 1
  or 3x = 30.
  

  Now we just have a simple one step equation.

• Step 3. Divide by 3 on both sides.
3x / 3
=
30 / 3
• Step 4. Simplify and rewrite the equation.
x = 10.

The very last step is to verify that you got the correct answer.

• Step 5. Plug 10 in for x.
3 / 2
(10) - 5 = 10

30 / 2
- 5 = 10
15 - 5 = 10
10 = 10 ✔