1. Involving One Operation
Lesson 2-1
Solving Equations
Involving One Operation

2. Standards California Preparation for 5.0
Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable, and provide justification for each step.
Also covered:
2.0

3. The two sides on a balanced scale must be equal to each other
A Question of Balance
The two sides on a balanced scale must be equal to each other
E =
E =
What does the Egg weigh?

4. What happens if we change one of the sides of a balanced equation?
A Question of Balance
What happens if we change one of the sides of a balanced equation?
8 + 3
+ 1 11
Then it is not balanced!

5. We need to make the same change to the other side!
A Question of Balance
What happens if we change one of the sides of a balanced equation? 11
+ 1
Then it is not balanced!
We need to make the same change to the other side!

6. A Question of Balance 8 + 3 + 1 11 + 1
What happens if we change one of the sides of a balanced equation?
11 + 1
We need to make the same change to the other side!
We need to make the same
change to the other side!

7. To solve an equation means to find every number that makes the equation true.
We do this by adding or subtracting to each side of the equation … but always keep it balanced!

8. Multiply by x. Divide by x.
To find solutions, perform inverse operations until you have isolated the variable. A variable is isolated when it appears by itself on one side of an equation, and not at all on the other side.
Inverse Operations
Add x Subtract x.
Multiply by x Divide by x.

9. Let’s go back to the balance
In the equation,
7 added to a number gives 15…
Solving the equation means, finding the value of the variable that makes the equation true.
Let’s go back to the balance

10. x +7
- 7 15
- 7
Subtract 7 from both sides
Simplify both sides

11. Subtraction Property of Equalityx 8
Subtract 7 from both sides
Now we know the value of x
Simplify both sides

12. Subtraction Property of Equality
Subtracting the same number from each side of an equation produces an equivalent equation x 8
So the solution goes like this…
x + 7 = 15
x + 7 – 7 = 15 – 7
Subtract 7 from both sides
Simplify both sides
x = 8
Now we know the value of x

13. Solving equations using Addition and Subtraction

14. The Box Method Let’s try to solve backwards… x – 11 = 13 - 11 x 13
+ 11 24 13

15. You Try Solve the equation. y – 8 = 24 + 8 + 8 y = 32 - 8 y 24 + 8 32
Since 8 is subtracted from y, add 8 to both sides to undo the subtraction.
y – 8 = 24
- 8 y
The solution set is {32}. 24
y = 32
+ 8
Check
y – 8 = 24 32 24
32 – 
To check your solution, substitute 32 for y in the original equation.

16. Solve the equation. Check your answer.
You try
Solve the equation. Check your answer.
Since 6 is subtracted from k, add 6 to both sides to undo the subtraction.
–6 = k – 6
- 6
The solution set is {0}. k -6
0 = k
+ 6
Check
–6 = k – 6 -6
– – 6
– –6 
To check your solution, substitute 0 for k in the original equation.

17. Solve the equation. Check your answer.
Once more
Solve the equation. Check your answer.
Since 6 is added to t, subtract 6 from both sides to undo the addition.
6 + t = 14
– – 6 +6
The solution set is {8}.
t = 8 t 14
Check
6 + t = 14
- 6 8 14 
To check your solution, substitute 8 for t in the original equation.

18. Now you try x + 4 = 10 2) 12 = x + 8 y + 7.1 = 1.2 4) 5.3 = y – 2
5) n – 6 = ) -3 = w + 1 -4 -4 -8 -8
x = 6
4 = x +2
- 7.1
- 7.1 +2
y = -5.9
7.3 = y +6 +6 -1 -1
-4 = w
n = -3

19. Solving equations using Multiplication and Division

20. Solve each equation. Check your answer.
Since y is multiplied by 0.5, divide both sides by 0.5 to undo the multiplication.
5y = –85
5y = –85
The solution set is {–20}. x5 y
-85
y = -17 ÷5
-17
-85

21. ÷ 5 p 10 x 5 50 10 Solve each equation. Check your answer.
Since p is divided by 5, multiply both sides by 5 to undo the division.
The solution set is {50}.
÷ 5
p = 50 p 10
Check
x 5 50 10
To check your solution, substitute 50 for p in the original equation.

22. In some equations, the solution is obvious.
x – 7 = 12
5n = 35
x = 19
n = 7
= 3
20 + h = 41
h = 21
c = 24
We can simply work the operation backwards in our head to get the answer.

23. But in other equations, the solution is not so obvious.
We have to know what operation(s) must be done to solve it, and work it out carefully.

24. But in other equations, the solution is not so obvious.
You have to do the inverse operation to both sides to get the variable by itself
But in other equations, the solution is not so obvious.
The inverse operation of subtraction is addition
The inverse operation of addition is subtraction
Addition property of equality
Subtraction property of equality
The inverse operation of division is multiplication
The inverse operation of multiplication is division
Multiplication property of equality
Division property of equality

25. Solve the equations below…
4x = ) x 3
4) y 2
5) 6n = ) = 3w
11.2 =
x = 4
x = x 3
12 =
16 =
x = 36
x = 7
w = - 7

26. Solving Equations by Using Opposites or Reciprocals
Solve each equation.
The reciprocal of is Since w is multiplied by multiply both sides by .
The solution set is {–24}.

27. Lesson Quiz Solve each equation. 1. r – 4 = –8 3. 2. 5.
6. This year a high school had 578 sophomores enrolled. This is 89 less than the number enrolled last year. Write and solve an equation to find the number of sophomores enrolled last year. –4 2.
2.8
4. 8y = 4 40
s – 89 = 578; s = 667