1. 2.4 Solving Equations with Variables on Both Sides:
No Solution: an equation has no solution
if there is no value to make the equation TRUE.
Infinitely Many Solutions: An equation that is true for any and every possible value.
Identity: an equation that has infinitely many solutions.

2. GOAL:

3. Ex: Solve 2x – 3 = x+5 + 3 +3 2x = x+8 -x -x x = 8
We can find the solution to equations that have variables on both sides of the equal sign by using inverse operations and moving the smallest coefficient to the other side of the equal sign:
Ex: Solve 2x – 3 = x+5
Isolate the variable with biggest coefficient
2x = x+8
-x x
x = 8

4. Ex: Solve 2x – 3 = x+5
2x = x+8
-x x
x = 8
Check: ( ) -3 = ( ) (8)-3 = (8) – 3 = =13

5. REAL-WORLD:
A dance studio charges $50 sign-up fee and $65 per day to take all dance classes. Another studio charges a $90 sign-up fee and only $45 per day to take all classes. For what number of days is the cost of the two dance studios the same?

6. Equal  65x + 50 = 45x + 90 SOLUTION: Using the given info we have:
Studio 1  $50 sign-up fee  +50
Studio 1  $65 per day  65x
Studio 2  $90 sign-up fee  +90
Studio 2  $45 per day  45x
Equal  x + 50 = 45x + 90

7.  65x + 50 = 45x + 90
65x + 16 = 45x + 90
Like terms on same side of equ.
-45x x
20x + 16 = 90
Inverse of add
20x = 74
Inverse of multiply
20x /20= 74/20
x = 4 days

8. YOU TRY IT:
What is the solution of
5X – 1 = X + 15?

9. Solving equations with Distributive Property:
To solve equations that include distributive property, we must distribute first, then isolate:
Ex: What is the solution of
4(2y+1)=2(y -13)?

10. y = - 5 Solution: 4(2y+1)= 2(y-13) Distributive 4 and 2
Multiplication
Inverse of +4 (addition)
– –4
8y = 2y - 30
-2y y
Move the smallest2y
6y = -30
_____ ____
Inverse of multiplication
y = - 5

11. Check:
4(2y+1) = 2(y-13)
4(2( )+1) = 2(( )-13)
4(2(-5)+1) = 2((-5)-13)
4(-10+1) = 2(-5-13)
4(-9) = 2(-18)
- 36 = - 36

12. YOU TRY IT:
What is the solution of:
𝟕 𝟒−𝐚 =𝟑(𝐚−𝟒)?

13. y = - 5 Solution: 4(2y+1)= 2(y-13) Distributive 4 and 2
Multiplication
Inverse of +4 (addition)
– –4
8y = 2y - 30
-2y y
Move the smallest2y
6y = -30
_____ ____
Inverse of multiplication
y = - 5

14. Note:
Whenever we solve for an equation for a given variable we might get ONE solution, Infinitely many solutions or NO solutions at all.

15. ONE SOLUTION:
What is the solution of
3(5b-2)= 6 +12b?

16. Solution: 3(5b-5)= -6+12b Distributive 3 3(5b) -3(5)= –6+12b
Multiplication
Inverse of subtraction
15b = 12b + 15
- 12b -12b
Inverse of multiplication
3b = 15
3b/3 = 15/3
Thus b = 5 is our one solution.

17. YOU TRY IT:
What is the solution of:
2a + 3 = a + 10?

18. INFINITELY MANY SOLUTIONS:
What is the solution of
3(4b-2)= b?

19. Solution: 3(4b-2)= -6+12b Distributive 3 3(4b) -3(2)= –6+12b
Multiplication
Inverse of subtraction
12b = 12b
- 12b -12b
Inverse of multiplication
0 = 0
Since 0 will always be 0, we have infinite solutions.

20. YOU TRY IT:
What is the solution of:
2a + 3 = ½ (6+4a)?

21. NO Solution:
What is the solution of
2x + 7= -(3 - 2x)?

22. Solution: 2x + 7 = -1(3 – 2x) Distributive -1 2x + 7 = –3 + 2x
Multiplication
Inverse of subtraction
2x +10 = 2x
Move the smallest2x
-2x x
10 = 0
Since 10 will never equals 0, there is NO solution.

23. YOU TRY IT:
What is the solution of:
3d + 4 =2 + 3d – ½ ?

24. VIDEOS: Multi-Step Equations Multi-Step

25. Problems: As many as it takes you to master the concept.
CLASS WORK:
Pages: 105 – 106
Problems: As many as it takes you to master the concept.