This document provides examples for solving two-step equations and formulas. It begins by defining a two-step equation as an equation that requires two steps to solve. It then shows worked examples of solving two-step equations by adding or subtracting the same quantity to both sides. The document also demonstrates solving formulas for a given variable and checking solutions. Key steps include isolating the variable of interest and performing inverse operations.

1. SECTION 3-4
Equations with Two or More Operations

2. ESSENTIAL QUESTION
How do you solve two-step equations and formulas?
Where you’ll see this:
Physics, mechanics, sports, modeling

3. VOCABULARY
1. Two-step Equation:

4. VOCABULARY
1. Two-step Equation: An equation that requires two steps to
solve

5. EXAMPLE 1
Solve.
3x − 7 = 14

6. EXAMPLE 1
Solve.
3x − 7 = 14
Step 1:

7. EXAMPLE 1
Solve.
3x − 7 = 14
Step 1: +7 +7

8. EXAMPLE 1
Solve.
3x − 7 = 14
Step 1: +7 +7
3x = 21

9. EXAMPLE 1
Solve.
3x − 7 = 14
Step 1: +7 +7
3x = 21
Step 2:

10. EXAMPLE 1
Solve.
3x − 7 = 14
Step 1: +7 +7
3x = 21
Step 2: 3 3

11. EXAMPLE 1
Solve.
3x − 7 = 14
Step 1: +7 +7
3x = 21
Step 2: 3 3
x=7

12. EXAMPLE 2
Practice these equations. Final solutions will be shown.
a. 4x + 9 = 33 b. 8 − 2 y = 26
4
c. 6t − (−5) = 41 d. 3 + g = 9
3

13. EXAMPLE 2
Practice these equations. Final solutions will be shown.
a. 4x + 9 = 33 b. 8 − 2 y = 26
x=6
4
c. 6t − (−5) = 41 d. 3 + g = 9
3

14. EXAMPLE 2
Practice these equations. Final solutions will be shown.
a. 4x + 9 = 33 b. 8 − 2 y = 26
x=6 y = −9
4
c. 6t − (−5) = 41 d. 3 + g = 9
3

15. EXAMPLE 2
Practice these equations. Final solutions will be shown.
a. 4x + 9 = 33 b. 8 − 2 y = 26
x=6 y = −9
4
c. 6t − (−5) = 41 d. 3 + g = 9
3
t=6

16. EXAMPLE 2
Practice these equations. Final solutions will be shown.
a. 4x + 9 = 33 b. 8 − 2 y = 26
x=6 y = −9
4
c. 6t − (−5) = 41 d. 3 + g = 9
3
t=6 9
g=
2

17. EXAMPLE 3
Solve the equation and check the solution.
3
a. − 3( x − 5) = 5( x − )
5

18. EXAMPLE 3
Solve the equation and check the solution.
3
a. − 3( x − 5) = 5( x − )
5

19. EXAMPLE 3
Solve the equation and check the solution.
3
a. − 3( x − 5) = 5( x − )
5
−3x + 15 = 5x − 3

20. EXAMPLE 3
Solve the equation and check the solution.
3
a. − 3( x − 5) = 5( x − )
5
−3x + 15 = 5x − 3
+3x +3x

21. EXAMPLE 3
Solve the equation and check the solution.
3
a. − 3( x − 5) = 5( x − )
5
−3x + 15 = 5x − 3
+3x +3 +3x +3

22. EXAMPLE 3
Solve the equation and check the solution.
3
a. − 3( x − 5) = 5( x − )
5
−3x + 15 = 5x − 3
+3x +3 +3x +3
18 = 8x

23. EXAMPLE 3
Solve the equation and check the solution.
3
a. − 3( x − 5) = 5( x − )
5
−3x + 15 = 5x − 3
+3x +3 +3x +3
18 = 8x
8 8

24. EXAMPLE 3
Solve the equation and check the solution.
3
a. − 3( x − 5) = 5( x − )
5
−3x + 15 = 5x − 3
+3x +3 +3x +3
18 = 8x
8 8
18
x=
8

25. EXAMPLE 3
Solve the equation and check the solution.
3
a. − 3( x − 5) = 5( x − )
5
−3x + 15 = 5x − 3
+3x +3 +3x +3
18 = 8x
8 8
18 9
x= =
8 4

26. EXAMPLE 3
Solve the equation and check the solution.
b. 4 p + 5 p − 13 = 14

27. EXAMPLE 3
Solve the equation and check the solution.
b. 4 p + 5 p − 13 = 14

28. EXAMPLE 3
Solve the equation and check the solution.
b. 4 p + 5 p − 13 = 14
9p − 13 = 14

29. EXAMPLE 3
Solve the equation and check the solution.
b. 4 p + 5 p − 13 = 14
9p − 13 = 14
+13 +13

30. EXAMPLE 3
Solve the equation and check the solution.
b. 4 p + 5 p − 13 = 14
9p − 13 = 14
+13 +13
9 p = 27

31. EXAMPLE 3
Solve the equation and check the solution.
b. 4 p + 5 p − 13 = 14
9p − 13 = 14
+13 +13
9 p = 27
9 9

32. EXAMPLE 3
Solve the equation and check the solution.
b. 4 p + 5 p − 13 = 14
9p − 13 = 14
+13 +13
9 p = 27
9 9
p=3

33. EXAMPLE 4
Solve the formula for t.
I = prt

34. EXAMPLE 4
Solve the formula for t.
I = prt
pr pr

35. EXAMPLE 4
Solve the formula for t.
I = prt
pr pr
I
t=
pr

36. EXAMPLE 5
Solve the formula for area of a triangle for b, then find the
base for the following situations.
a. Height = 4 m, Area = 12 m2

37. EXAMPLE 5
Solve the formula for area of a triangle for b, then find the
base for the following situations.
a. Height = 4 m, Area = 12 m2
1
A = bh
2

38. EXAMPLE 5
Solve the formula for area of a triangle for b, then find the
base for the following situations.
a. Height = 4 m, Area = 12 m2
1
A = bh
2
2 1 2
h ( A) = ( 2 bh) h

39. EXAMPLE 5
Solve the formula for area of a triangle for b, then find the
base for the following situations.
a. Height = 4 m, Area = 12 m2
1
A = bh
2
2 1 2
h ( A) = ( 2 bh) h
2A
=b
h

40. EXAMPLE 5
Solve the formula for area of a triangle for b, then find the
base for the following situations.
a. Height = 4 m, Area = 12 m2
1 2A
A = bh
2 b=
2 1 2 h
h ( A) = ( 2 bh) h
2A
=b
h

41. EXAMPLE 5
Solve the formula for area of a triangle for b, then find the
base for the following situations.
a. Height = 4 m, Area = 12 m2
1 2A
A = bh
2 b=
2 1 2 h
h ( A) = ( 2 bh) h
2(12)
2A b=
=b 4
h

42. EXAMPLE 5
Solve the formula for area of a triangle for b, then find the
base for the following situations.
a. Height = 4 m, Area = 12 m2
1 2A
A = bh
2 b=
2 1 2 h
h ( A) = ( 2 bh) h
2(12) 24
2A b= =
=b 4 4
h

43. EXAMPLE 5
Solve the formula for area of a triangle for b, then find the
base for the following situations.
a. Height = 4 m, Area = 12 m2
1 2A
A = bh
2 b=
2 1 2 h
h ( A) = ( 2 bh) h
2(12) 24
2A b= = =6
=b 4 4
h

44. EXAMPLE 5
Solve the formula for area of a triangle for b, then find the
base for the following situations.
a. Height = 4 m, Area = 12 m2
1 2A
A = bh
2 b=
2 1 2 h
h ( A) = ( 2 bh) h
2(12) 24
2A b= = =6m
=b 4 4
h

45. EXAMPLE 5
Solve the formula for area of a triangle for b, then find the
base for the following situations.
b. Height = 15 in, Area = 18 in2

46. EXAMPLE 5
Solve the formula for area of a triangle for b, then find the
base for the following situations.
b. Height = 15 in, Area = 18 in2
2A
b=
h

47. EXAMPLE 5
Solve the formula for area of a triangle for b, then find the
base for the following situations.
b. Height = 15 in, Area = 18 in2
2A
b=
h
2(18)
b=
15

48. EXAMPLE 5
Solve the formula for area of a triangle for b, then find the
base for the following situations.
b. Height = 15 in, Area = 18 in2
2A
b=
h
2(18) 36
b= =
15 15

49. EXAMPLE 5
Solve the formula for area of a triangle for b, then find the
base for the following situations.
b. Height = 15 in, Area = 18 in2
2A
b=
h
2(18) 36 12
b= = =
15 15 5

50. EXAMPLE 5
Solve the formula for area of a triangle for b, then find the
base for the following situations.
b. Height = 15 in, Area = 18 in2
2A
b=
h
2(18) 36 12 2
b= = = =25
15 15 5

51. EXAMPLE 5
Solve the formula for area of a triangle for b, then find the
base for the following situations.
b. Height = 15 in, Area = 18 in2
2A
b=
h
2(18) 36 12 2
b= = = = 2 5 in
15 15 5

52. HOMEWORK

53. HOMEWORK
p. 118 #1-11 all, 13-39 odd
“I just never let anything bother me, man. I know myself
really well. Nobody’s opinion of me can shake my opinion
of myself.” -Ruben Studdard