Math 8

CHAPTER 2-1: Equations in One Variable

REVIEW OF SIMPLE EQUATIONS! One and Two Step!

GOAL? Determine the value of the variable

HOW? Isolate the variable (get it alone on one side of equation)

WHAT DO I DO? Use inverse (opposite) operations to "get rid" of everything on the side with the variable

WHAT SHOULD MY FOCUS BE WHEN EQUATIONS GET COMPLICATED?

Always focus on the variable(s) first!!!!!!!

We'll be meeting our old BFFs from 7th grade!

EQUATION BALANCING PROPERTIES OF EQUALITY:

Whatever YOU DO TO BALANCE an equation,

that operation is the property of equality that was used.

If you have x + 3 = 10, you used the SUBTRACTION PROPERTY OF EQUALITY because you need to SUBTRACT 3 from each side equally.

If you have x - 3 = 10, you used the ADDITION PROPERTY OF EQUALITY because you need to ADD 3 from each side equally.

If you have 3x = 10, you used the DIVISION PROPERTY OF EQUALITY because you need to DIVIDE each side equally by 3.

If you have x/3 = 10, you used the MULTIPLICATION PROPERTY OF EQUALITY because you need to MULTIPLY each side equally by 3.

SOMETIMES, WE SAY THERE ARE ONLY 2 BALANCING PROPERTIES OF EQUALITY

CAN YOU GUESS WHICH 2 ARE "DROPPED OUT"?

Since we say we never subtract and we really never divide, it's those 2.

GOING BACK TO OUR PREVIOUS EXAMPLES:

If you have x + 3 = 10, you could say that we ADDED -3 to each side equally; therefore, we used the ADDITION (not subtraction) PROPERTY.

If you have 3x = 10, you could say that we MULTIPLIED each side equally by 1/3; therefore, we used the MULTIPLICATION (not division) PROPERTY.

(We always multiply by the MULTIPLICATIVE INVERSE).

Special type of one-step equation are those where the VARIABLE IS NEGATIVE.

Remember: You’re solving for the POSITIVE VARIABLE.

There are a couple of ways to do this.

DID YOU KNOW THAT YOU CAN MOVE A NEGATIVE SIGN

IN 3 DIFFERENT PLACES ON ANY FRACTION????

So if you see a negative sign on a variable in a fraction, you can move the negative sign!

We use the MULTIPLICATIVE INVERSE PROPERTY to balance the equation and ISOLATE the variable on one side.

The multiplicative inverse of a number is its reciprocal.

The multiplicative inverse of ¾ is 4/3.

If the fraction is NEGATIVE, so is its multiplicative inverse:

Example: If you have -3 ½ , first make it improper: -7/2, and then its multiplicative inverse (reciprocal) is -2/7.

MULTIPLY BY THE RECIPROCAL ON BOTH SIDES TO SOLVE A ONE-STEP EQUATION WHERE THE COEFFICIENT IS A FRACTION.

FOR DECIMALS, SIMPLY DIVIDE EACH SIDE BY THE DECIMAL AND DO THE DIVISION USING YOUR UNDERSTANDING OF LONG DIVISION WITH DECIMALS.

I’ll also show you how to get rid of the decimals as a different approach.

If you’d like to try this approach for decimals:

Multiply both sides of the equation by a POWER OF 10 big enough to get rid of the all the decimals (make them all integers).

Example: 3.45x = .005

You would need to multiply both sides by 1000 so that both sides would no longer have decimals:

1000(3.45x) = 1000(.005)

3450x = 5

Now you would divide by 3450 on both sides.

Note: You’ll still get a decimal answer, but while you’re solving you don’t have decimals.

This works really well when all the terms have the same place value!

If the coefficient is a fraction, you’ll get the answer in ONE-STEP if you use the MULTIPLICATIVE INVERSE PROPERTY and multiply both sides equally by the RECIPROCAL of the coefficient.

Example:

¾ x = 15

(4/3)(¾ x) = (15)(4/3)

x = 20

MAKE SURE YOU ALWAYS CROSS CANCEL if possible!!!