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.

Algebra Dougie

One more, this time to Teach Me How to Dougie.

Video Link

Repeat It (Algebra)

MJ's classic Beat It with different lyrics that apply to what we're learning.

Have a little fun!

Video Link

Something Funny...

And now for something completely different!

Brought to you from the deep dark recesses of youtube:  Hint: Stick with it and you'll hear Thrift Shop.

Video Link

Graphing Linear Equations

Most of you know what a linear equation is.  You've not heard it called this, but you know what it is.  All the sequences we've had so far form linear equations.  So what are some things you can take away from what we've done with sequences?

1.  All linear equations have a constant rate of change.  As in, when x goes up by 1, y goes up by the same number each time, whether it's up by 2, 3, 5, etc.

We call this rate of change m.  In equation form, it's the number that is multiplied by x (remember inverses mean that it could be the number you're dividing by, too!).

These equations also have a constant number where x crosses the y-axis, called the y-intercept.  You find this by setting x to zero.  Then when you multiply by the change, the change falls out, leaving the intercept.  This intercept is called b.

A linear equation has a few different forms, but the most workable form is called the slope-intercept form.  This means we are solving for y by doing stuff to x.  So y = m * x + b

We can then graph the line on a coordinate plane based on using a process table for y = m * x + b.

You can see that demonstrated here:

Properties of Equality

The properties of equality are basic algebraic facts that help us solve equations.  We are already using these, we just haven't put names to them yet.  So in this video this teacher puts the name with the concept.  You will cover this again in 9th grade and refresh it in 10th and 11th grade.  So you might as well learn it now so that it's faster in 9th grade and an old concept in 11th grade.

To see this concept in more detail, watch this video.

The major takeaway are the properties of equality for the operations.  Used in conjunction with the additive and multiplicative identities and inverses, we can solve for variables.

Here is the list of properties:

Additive identity: a + 0 = a (or 4 + 0 = 4, etc)

Multiplicative identity:  a * 1 = a  (or 4 * 1 = 4, etc)

Additive inverse: a + -a = 0 (or 4 + -4 = 0, etc)

Multiplicative inverse a * 1/a = 1 (or 4 * 1/4 = 1, etc)

Addition property of equality: if a = b, then a + c = b + c  (if 4 = 4, then 4 + 2 = 4 + 2)

Subtraction property of equality: if a = b, then a - c = b - c (if 4 = 4, then 4 - 2 =  4 - 2)

Multiplication property of equality: if a = b, then a * c = b * c (if 4 = 4, then 4 * 2 = 4 * 2)

Division property of equality: if a = b, then a / c = b / c (if 4 = 4, then 4 / 2 = 4 / 2)

Using these properties, we can solve any equation with 1 variable.  We can simplify any equation with 2 variables.  Understanding these properties will help you write equations for word problems, or pick the correct equation out of a list.

Properties of Arithmetic

I'm sure some of you have wondered "how does Mr. V. know all these tricks?"  One reason is because I have learned this set of arithmetic properties here:

http://www.coolmath.com/prealgebra/06-properties/

Many of you are already using these without knowing it.  The multiplicative inverse property is used in solving proportions through butterfly method.

The additive inverse property is used to solve for x in linear relationships when y is constant.

I just like the way the concepts are presented at coolmath. I hope you do too.

Oh, and if you don't learn them now, you will continue to struggle and will probably have trouble in the rest of your math career through high school and possibly beyond.

Linear Patterns

Here's a video explaining how to find a pattern based on a sequence.

Video Link

Most of you are fairly accomplished at this method.  What you have been having trouble at is understanding how to write the rule.  Mr. Khan shows a way to understand the rule in this video.

Video Link

Here's a more complicated example that actually gets Mr. Khan going into Algebra 1!

Video Link

Here's one more example.

Video Link

Hope these help you prepare.