1. LESSON 1.11 SOLVING EQUATIONS

2. Review of Terms
A variable is a letter which represents an unknown number. Any letter can be used as a variable.
An algebraic expression contains at least one variable.
Examples: a, x+5, 3y – 2z
A verbal expression uses words to translate algebraic expressions.
Example: “The sum of a number and 3” represents “n+3.”
An equation is a sentence that states that two mathematical expressions are equal.
Example: 2x-16=18

3. Steps to Solving Equations
Simplify each side of the equation, if needed, by distributing or combining like terms.
Move variable(s) to one side of the equation by using the opposite operation.
Isolate the variable by applying the opposite operation to each side.
First, use the opposite operation of addition or subtraction.
Second, use the opposite operation of multiplication or division.
Check your answer.

4. Examples “y” is the variable.
Add 6 to each side to isolate the variable.
Now divide both sides by 3.
The answer is 5.
Check the answer by substituting it into the original equation.

5. ONE-STEP EQUATIONS

6. Objective The student will be able to:
solve equations using addition and subtraction.

7. Think of this equation as a balance scale.
1) Solve r + 16 = -7
Think of this equation as a balance scale.
r + 16 -7 =
Whatever you do to one side has to be done to the other to keep it balanced!

8. 1) Solve r + 16 = -7 To solve, you must get the variable by itself.
What number is on the same side as r? 16
To get r by itself, we must undo the “add 16”. What is the opposite of addition?
Subtract 16

9. 1) Solve r + 16 = -7
r = -23
= -7
Draw “the river” to separate the equation into 2 sides
Subtract 16 from both sides
Simplify vertically
Check your answer by substituting your answer back into the problem

10. 2) Solve x + 2 = -3 Get the variable by itself. What is your first step?
Add 2 to both sides
Subtract 2 from both sides
Add 3 to both sides
Subtract 3 from both sides
Answer Now

11. 2) Solve x + 2 = -3
x = -5
= -3
Draw “the river” to separate the equation into 2 sides
Subtract 2 from both sides
Simplify vertically
Check your answer by substituting your answer back into the problem
On homework and tests, be sure to check your work!!
There is no reason why you should miss a problem!

12. 3) Solve 8 = m - 3
m = 5
m = 11
m = 24
m = 8/3
Answer Now

13. 3) Solve 8 = m - 3
11= m
8 =
Draw “the river” to separate the equation into 2 sides
Add 3 to both sides
Simplify vertically
Check your answer by substituting your answer back into the problem

14. When solving equations, we want to eliminate double signs.
y + (-3) = 8
is rewritten as
y – 3 = 8
p – (-5) = 6
p + 5 = 6
As a general rule, replace “+ (- )” with “–” and “– (- )” with “+”. This will make things less confusing in the future!

15. 4) Solve y + (-3) = 7 y – 3 = 7 + 3 +3 y = 10 10 + (-3) = 7
y = 10
10 + (-3) = 7
Draw “the river” to separate the equation into 2 sides
Eliminate the double sign
Add 3 to both sides
Simplify vertically
Check your answer by substituting your answer back into the problem

16. 5) Solve. -x - (-2) = 1 -x + 2 = 1 - 2 - 2 -x = -1 x = 1 -(1) + 2 = 1
Draw “the river” to separate the equation into 2 sides
Eliminate the double sign
Subtract 2 from both sides
Simplify vertically
We haven’t gotten x by itself. If we read this aloud, it is “the opposite of x equals -1”. What would x be equal?
Check your answer
-x + 2 = 1
-x = -1
x = 1
-(1) + 2 = 1

17. Solve -y – (-3) = 7
y = 10
y = 4
y = -10
y = -4
Answer Now

18. Objective The student will be able to:
solve equations using multiplication and division.

19. To get the variable by itself, which number needs to be moved?
1) Solve. -5t = 60
To get the variable by itself, which number needs to be moved? -5
To move the -5, you have to do the opposite operation. What operation will we use?
division

20. 1) Solve -5t = 60
-5 -5
t = -12
-5(-12) = 60
Draw “the river” to separate the equation into 2 sides
Divide both sides by -5
Simplify
Check your answer

21. 2) Solve 15 = 6n 6 6 2.5 = n 15 = 6(2.5) Draw “the river”
6 6
2.5 = n
15 = 6(2.5)
Draw “the river”
Divide both sides by 6
Simplify
Check your answer

22. 3) Solve -3v = -129
v = -126
v = -43
v = 43
v = 126
Answer Now

23. 4) Solve You don’t like fractions? Let’s get rid of them! 
“Clear the fraction”
by multiplying both sides of the equation by the denominator.

24. 4) Solve 4 · · 4 x = -48 Draw “the river”
Clear the fraction – multiply both sides by 4
Simplify
Check your answer
4 · · 4
x = -48

25. 5) Solve = 18 3 · = 18 · 3 2x = 54 2 2 x = 27 Draw “the river”
3 · = 18 · 3
2x = 54
x = 27
Draw “the river”
Clear the fraction – multiply both sides by 3
Simplify
Divide both sides by 2
Check your answer

26. 6) Which step clears the fraction in
Multiply by 3
Multiply by 5
Multiply by -12
Multiply by -5
Answer Now

27. 7) Solve
b = -56
b = -14
b = 14
b = 56
Answer Now

28. TWO-STEP EQUATIONS

29. Objective The student will be able to:
solve two-step equations.

30. To solve two-step equations, undo the operations by working backwards.
Recall the order of operations as you answer these questions.
Dividing by 2
Subtracting 3
Example:
Ask yourself,
What is the first thing we are doing to x?
What is the second thing?
To undo these steps, do the opposite operations in opposite order.

31. Use a DO-UNDO chart as a shortcut to answering the questions
Use a DO-UNDO chart as a shortcut to answering the questions. In the table, write the opposite operations in the opposite order
DO UNDO ÷2 -3
Follow the steps in the ‘undo’ column to isolate the variable.
Draw “the river”
Add 3 to both sides
Simplify
Clear the fraction -Multiply both sides by 2
Check your answer +3
· 2
= - 4
x = -8
-4 – 3 = -7
2 · · 2

32. 1) Solve 2x - 1 = -3 + 1 + 1 2x = -2 2 2 x = -1 2(-1) - 1 = -3
D U
1) Solve 2x - 1 = -3
· 2
- 1
+ 1
÷ 2
2x = -2
x = -1
2(-1) - 1 = -3
-2 – 1 = -3
Add 1 to both sides
Simplify
Divide both sides by 2
Check your answer

33. 2) Solve + 4 + 4 3 · · 3 x = 36 12 – 4 = 8 D U Add 4 to both sides
÷ 3
- 4
+ 4
· 3
3 · · 3
x = 36
12 – 4 = 8
Add 4 to both sides
Simplify
Clear the fraction - Multiply both sides by 3
Check your answer

34. Solving equations with brackets:
2 (x + 3) = x + 11
Multiply out the bracket
2x + 6 = x + 11
Subtract x from each side
x + 6 = 11
Subtract 6 from each side
x = 5

35. WITH VARIABLES ON BOTH SIDES
EQUATIONS
WITH VARIABLES ON BOTH SIDES

36. Objectives The student will be able to:
1. solve equations with variables on both sides.
2. solve equations containing grouping symbols.

37. 1) Solve. 3x + 2 = 4x - 1
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.

38. 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 =
Subtract 3x from both sides
Simplify
Add 1 to both sides
Check your answer

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

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

41. 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
Distribute
Subtract 21 from both sides
Simplify
Divide both sides by -7
Check your answer

42. -2x -2x 5 = -3 This is never true! No solutions
Special Case # ) 2x + 5 = 2x - 3
-2x x
5 = -3
This is never true!
No solutions
Subtract 2x from both sides
Simplify

43. Infinite solutions or identity
Special Case # ) 3(x + 1) - 5 = 3x - 2
3x + 3 – 5 = 3x - 2
3x - 2 = 3x – 2
-3x x
-2 = -2
This is always true!
Infinite solutions or identity
Distribute
Combine like terms
Subtract 3x from both sides
Simplify

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

45. Objectives The student will be able to:
1. solve equations with variables on both sides.
2. solve equations containing grouping symbols.

46. 1) Solve. 3x + 2 = 4x - 1
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.

47. 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

48. 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

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

50. 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

51. 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

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

53. 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

54. -2x -2x 5 = -3 This is never true! No solutions
Special Case # ) 2x + 5 = 2x - 3
-2x x
5 = -3
This is never true!
No solutions
Draw “the river”
Subtract 2x from both sides
Simplify

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

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

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