Solving 2-Step Equations

Solving equations needs the properties of basic operations and equality.  A quick review can be found here.

These problems typically look like:

Things to note: Any number next to the x is being multiplied by x.  Watch for addition and subtraction.  One of the rules about adding variables is that if you try to divide first, you have to divide anything you're adding or subtracting, so it is typically a good idea to remove that part first.  In this case, 20 is being added to 3 times a number to get 41.  So to isolate the number, we would need to understand and use the identity property of addition and the inverse property of addition, which both have to do with adding or subtracting to get zero and then adding zero to get the same number.  So in order to get just 3 times a number, we need to have it as 3 times a number plus zero.  Using the inverse property, 20 + -20 = 0.  This gives us:

We must subtract 20 from both sides to keep equality.  If you could take some only from one side, then it would indicate that they weren't equal to begin with, which would make the problem wrong from the beginning.  This gives us:

Again to get the unknown number by itself, we need to use identity and inverse properties of multiplication.  This means finding a number that we can multiply by 3 to get one.  Some of you remember from learning fractions that any time you divide fractions, you can multiply by their reciprocal.  The reciprocal is the opposite part/whole ratio to your first fraction.You might also remember that any whole number can be written as a fraction by putting it over one.  So 3 becomes 3/1.  To make a fraction equal one, we multiply by its reciprocal.  3/1 * 1/3 = 1  Then we use the identity property 1 * x = x.  Like above, it would be wrong to only multiply one side by 1/3.  Thus:

This gives us x = 21/3 or:

There is the full example of how to work 2 step equations.  Common issues are as follows:

1.  Trying to multiply or divide first.  This can work but often leads to trouble.  In our example above, you would be able to solve it still, but the problem would be more difficult.  By increasing complexity, you increase your chance of a math error.  Multiplying first would lead to x + 20/3 = 41/3  If you have a strong dislike of fractions this could increase your distress.

2.  Picking the wrong number to turn to zero.  Subtracting 41 from both sides would give us x - 21 = 0.  Not the worst thing that can happen but it added an extra step where it was not needed.  Now you have to add 21 to both sides after.  Trying to multiply by zero is an exceedingly bad idea as that gives you 0x + 0 = 0.  x becomes all numbers, so instead of solving it, you've made it unsolvable!

3. Not making addition or subtraction zero:  This simply increases the number of steps needed and can increase your frustration level.  By subtracting 5 from both sides, you don't get very far with 3x + 15 = 36.  You could divide next but you'd still need to shift more to the other side.  Likewise with -10 from both sides.

4. Not making multiplication by 1.  Without the identity, you still have to deal with another step.  Just like not making addition/subtraction not zero, not multiplying by 1 gives you more steps .

5.  Not using the inverse properly.  The opposite of 20 is -20, not +20.  Likewise the reciprocal of 3 is 1/3, not 3 or some other number.

6.  Dividing by a number that is added or subtracted.  Like 3, this still creates problems.  In our problem, dividing by 20 would do very little.  3x/20 + 1 = 41/20 looks really ugly and difficult to solve.