1. 3.5 Solving Equations w/ Variables on Both Sides Goal: Solving Equations with variables on both sides

2. What!?! Variables on both sides!?! In some cases, we may have to solve equations with variables on both sides of the equal sign! Example 1: 2x - 4 = x - 2 Example 2 4x - 4 + 2x = 7x - 4 Example 3 3(x - 4) = 2(x - 4)+2x

3. So what can we do? We can undo or eliminate variables the exact same way we did constants. (numbers) What undoes a +2? A -2 will undo it or zero it out! What undoes a +2x? A -2x will undo it or zero it out!

4. Ex 1:Variables on Each Side Pick one of the variables to undo! Lets undo the -2x by adding 2x to each side! Solve

5. Example 1: Again? What if we undo the +7x instead? Solve No matter what variable we undo first we end with the same solution

6. Does it Matter what Variable we start with? Both methods have a solution of 4. It does not matter what variable you eliminate first, whatever you feel more comfortable with. If you eliminate the the variable with the smaller coefficient, your variable will stay positive thus reducing chance for error.

7. Example 2 does not simplify to an integer, so just reduce the fraction!! Eliminate the – 9y because that side has a constant also, where the 6y side only has a variable!

8. Try these! 2 = x-2 = x y = 3 -5 = x

9. Combine Like Terms First… Always look to simplify each expression before undoing any operations!

10. You try…

11. What happens if the variables disappear? What happens if the variables eliminate or undo each other?

12. Number of Solutions… So far all the equations we have solved have had one specific solution. We end up with x = some value. In some cases, the variables can totally eliminate each other from the equation!

13. Infinite Solution or Identity Notice the Variables eliminated each other… We are left with a balanced equation at the end. This means this is an identity and any value will be a solution to this equation. We say this equation has infinite solutions.

14. No Solutions Notice the Variables eliminated each other… We are left with an unbalanced equation at the end. This means this equation is impossible to solve. We say this equation has no solutions.

15. Remember… If the variables do not eliminate, there is one solution. If variables eliminate and equation is balanced there are infinite solutions. If variables eliminate and equation is unbalanced there is no solutions.

16. You try… Infinite solutions/ identity No Solutions X = -16X = 0