1. Solving Equations with Variables on Both Sides
Students will be able to solve equations with variables on both sides and solve equations containing grouping symbols.

2. Warm-Up #11 (3/6/2017) Is (3, 4) a solution to the equation 𝑥−𝑦=8
Solve for x. 10𝑥+5=2𝑥−15.

3. Homework (3/6/2017)
Worksheet: Equations with Infinite and No Solutions (ODD #, front and back)

4. You need to get the variables on one side of the equation
You need to get the variables on one side of the equation. It does not matter which variable you move. Try to move the one that will keep your variable positive.

5. 1) Solve 3x + 2 = 4x - 1 - 3x - 3x 2 = x - 1 + 1 + 1 3 = x
3 = x
3(3) + 2 = 4(3) - 1
9 + 2 =
Draw “the river”
Subtract 3x from both sides
Simplify
Add 1 to both sides
Check your answer

6. 2) Solve 8y - 9 = -3y + 2 + 3y + 3y 11y – 9 = 2 + 9 + 9 11y = 11 11 11
11y =
y = 1
8(1) - 9 = -3(1) + 2
Draw “the river”
Add 3y to both sides
Simplify
Add 9 to both sides
Divide both sides by 11
Check your answer

7. What is the value of x if 3 - 4x = 18 + x?-3 3
Answer Now

8. 3) Solve 4 = 7x - 3x 4 = 4x 4 4 1 = x 4 = 7(1) - 3(1) Draw “the river”
1 = x
4 = 7(1) - 3(1)
Draw “the river”
– Notice the variables are on the same side!
Combine like terms
Divide both sides by 4
Simplify
Check your answer

9. 4) Solve -7(x - 3) = -7 -7x + 21 = -7 - 21 - 21 -7x = -28 -7 -7 x = 4
-7x = -28
x = 4
-7(4 - 3) = -7
-7(1) = -7
Draw “the river”
Distribute
Subtract 21 from both sides
Simplify
Divide both sides by -7
Check your answer

10. What is the value of x if 3(x + 4) = 2(x - 1)?
-14
-13 13 14
Answer Now

11. 5) Solve 3 - 2x = 4x – 6 + 2x +2x 3 = 6x – 6 + 6 + 6 9 = 6x 6 6
Draw “the river”
Clear the fraction – multiply each term by the LCD
Simplify
Add 2x to both sides
Add 6 to both sides
Divide both sides by 6
Check your answer
3 - 2x = 4x – 6
+ 2x +2x
= 6x – 6
= 6x
or 1.5 = x

12. A system of linear equations is a set of two or more linear equations in the same variable. An example is shown below: y=𝑥+1 y=2𝑥− 7

13. A system of linear equations can have: 1. Exactly one solution
A solution to a system of equations is an ordered pair that satisfy all the equations in the system.
A system of linear equations can have:
1. Exactly one solution
2. No solutions
3. Infinitely many solutions

14. Infinite number of solutions
Consistent
Dependent
Inconsistent
One solution
Lines intersect
No solution
Lines are parallel
Infinite number of solutions
Coincide-Same line

15. ONE Solution There is only one answer that makes the equation true
Types of Solutions
ONE Solution
There is only one answer that makes the equation true
(EX: x=3 This is the answer)

16. Types of Solutions
NO Solution
There is no number that will satisfies both sides of the equation. Variables will cancel and a false statement is left.
(EX: 5=3 Answer: No solution)

17. MANY Solutions (Infinite)
Types of Solutions
Both lines are on top of each other
MANY Solutions (Infinite)
Any number substituted for the variable will make the equation true. All variables will cancel and a true statement will be left.
(EX: -2=-2 Answer: All real numbers)

18. Special Case #1 6) 2x + 5 = 2x - 3 -2x -2x 5 = -3 This is never true!
5 = -3
This is never true!
No solutions
Draw “the river”
Subtract 2x from both sides
Simplify

19. Special Case #2 7) 3(x + 1) - 5 = 3x - 2
-2 = -2
This is always true!
Infinite solutions or identity
Draw “the river”
Distribute
Combine like terms
Subtract 3x from both sides
Simplify

20. Special Case #3
3x + 4x – 4 = 10
7x – 4 = 10
7x =
7x = 14
x = 2
ONE SOLUTION

21. What is the value of x if -3 + 12x = 12x - 3?4
No solutions
Infinite solutions
Answer Now

22. Challenge! What is the value of x if -8(x + 1) + 3(x - 2) = -3x + 2?-2 2 8
Answer Now