CHAPTER 2-5: Multistep Equations
IDENTITY OR NO POSSIBLE SOLUTION EQUATIONS:
An identity equation is where ANY NUMBER can be substituted for the VARIABLE, the equation will be TRUE. What will happen is that while you’re balancing the equations, you will ultimately end up with the SAME EXACT EXPRESSION ON EACH SIDE of the equation.
You can keep going, but as soon as you have the same thing on both sides, you know you have an IDENTITY EQUATION. If you keep going, the variable will disappear on both sides of the equation, leaving you with a numeric equation (a number of both sides of the equal sign). That numeric equation will be TRUE.
Therefore, ALL REAL NUMBERS will work so we say there are INFINITELY MANY SOLUTIONS.
A no possible solution equation is one where no matter what number you substitute into the equation, the equation will be FALSE.
What will happen is that after you balance the equations, you will ultimately end up without any variable on either side, just like the Identity Equation, but this time, the numbers will be DIFFERENT (which can never be true).
THERE IS NO POSSIBLE SOLUTION OR THE NULL SET ∅
NOTICE SOMETHING ELSE ABOUT
SOLVING EQUATIONS IN GENERAL:
WHENEVER YOU HAVE THE SAME EXACT TERM WITH THE SAME SIGN ON DIFFERENT SIDES OF THE EQUATION, YOU CAN SIMPLY CROSS THEM OUT BECAUSE WHEN YOU USE THE ADDITIVE INVERSE PROPERTY ON BOTH SIDES TO BALANCE, BOTH TERMS WILL DROP OUT!
If there’s Distributive Property, generally, you will distribute first (there are times when you have the choice to divide first and I’ll show you that in class)
Simplify each side of the equation first...combine like terms on the same side!
Then use the ADDITIVE INVERSE PROPERTY to move variables to the other side of the equation so that all variables are on the same side.
Usually, we try to move the smaller coefficient to the larger because sometimes this avoids negative coefficients,
BUT that is not always the case, and you may move to whatever side you choose!
Here’s a way to remember to use the OPPOSITE SIGN when you move terms to the other side of the equation:
OPPOSITE SIDES
use the
OPPOSITE SIGN
However if like terms are on the same side of the equation, use the sign of the coefficient that you’re given and simply combine them by using integer rules:
SAME SIDE
use the
SAME SIGN
EXAMPLE OF COLLECTING TERMS FIRST:
Sometimes, you will have to COLLECT LIKE TERMS ON THE SAME SIDE OF THE EQUATION before balancing:
8y + 12 – (-2y) = -6
10y + 12 = -6
10y = -18
y = -18/10 = -9/5
TWO STEPS WITH DISTRIBUTIVE PROPERTY
Usually, you want to do DISTRIBUTE FIRST!
UNLESS THE FACTOR OUTSIDE THE ( ) CAN BE DIVIDED
OUT OF BOTH SIDES PERFECTLY! (which I will show you in class)
EXAMPLE:
5y - 2(2y + 8) = 16
5y - 4y - 16 = 16 [distribute]
y - 16 = 16 [collect like terms]
y = 32 [solve by adding 16 to both sides]
EXAMPLE WHEN YOU DON'T NEED TO DISTRIBUTE FIRST:
-3(4 + 3x) = -9
4 + 3x = 3 [Don't distribute! Divide by -3. The -3 goes into both sides perfectly!)
3x = -1 [Subtract 4 from both sides]
x = -1/3 [Divide both sides by 3]
EXAMPLE WHERE YOU NEED TO BOTH COMBINE ON THE SAME SIDE FIRST, AND THEN MOVE THE VARIABLES TO THE SAME SIDE:
3y - 10 - y = -10y + 12
The 3y and -y are on the SAME SIDE, so just use the SAME SIGNS as in the equation and combine them using integer rules:
2y - 10 = -10y + 12
Now the 2y and the -10y are on OPPOSITE SIDES of the equation so use the OPPOSITE SIGN to move the -10y to the left side:
2y - 10 = -10y + 12
+10 +10y
12y - 10 = 12
NOW DO THE TWO-STEP ;)
12y - 10 = 12
+ 10 +10
12y = 22
12 12
y = 11/6
