Algebra ( Period 1)

 SOLVING ONE-STEP EQUATIONS 2-2

REVIEW: EQUATION BALANCING PROPERTIES OF EQUALITY:

There are 4 of these.

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).

REVIEW OF SIMPLE EQUATIONS!
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!

IDENTITY PROPERTIES AND INVERSE PROPERTIES 

are also used to justify solving equations!

When you have a one-step equation such as x + 5 = 12, you ADD -5 (or just subtract 5) from each side equally. The reason you chose -5 is that it was the ADDITIVE INVERSE of 5.  The reason the +5 then "disappears" is due to the IDENTITY PROPERTY OF ADDITION. Since +5 + (-5) = 0, it's not necessary to bring down the 0 in the equation.

JUSTIFYING A SIMPLE ONE-STEP:

                                                   x + 5 = 12     GIVEN

                                                      - 5   -5     Subtraction Prop =

                                                        0            Additive Inverse Prop

                                                   x       =  7     Identity of Addition

FORMAL CHECK:
1. Rewrite original equation
2. Substitute your solution and question mark over the equal sign
3. Do the math and check it!

EXAMPLE FROM ABOVE:

                                      1.  Rewrite:                                  x + 5 = 12

                                                                                                    ?  

                                      2. Substitute your solution:        7 + 5 = 12

                                      3.  Do the math!                               12 = 12 √

Quick review of a couple of specific types of one-steps:

Do you remember from 7th grade how you balance an equation that has a fractional coefficient?

Multiply by the reciprocal (our BFF, the multiplicative inverse property ;)

Another 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, just MOVE IT to the number!

 

If you don’t move the negative sign first, BE CAREFUL because you’ll need to either multiply or divide by -1 at the very end to find POSITIVE y:

If there’s a negative on a variable and it’s not part of a fraction, you can multiply or divide both sides by

-1 AS I JUST SHOWED ABOVE

or you can just reason out the answer: