Properties of Equality
Topic 2.2.1
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Topic
2.2.1
Properties of Equality
California Standard:
What it means for you:
4.0 Students simplify expressions
before solving linear equations
and inequalities in one variable,
such as 3(2x – 5) + 4(x – 2) = 12.
You’ll solve one-step equations
using properties of equality.
Key words:
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expression
equation
variable
solve
isolate
equality
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Topic
2.2.1
Properties of Equality
Now it’s time to use the material on expressions you
learned in Section 2.1.
An equation contains two expressions, with an equals
sign in the middle to show that they’re equal.
For example: 2x – 3 = 4x + 5
In this Topic you’ll solve equations that involve addition,
subtraction, multiplication, and division.
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Topic
2.2.1
Properties of Equality
An Equation Shows That Two Expressions are Equal
An equation is a way of stating that two expressions have the same value.
This equation contains only numbers — there are no unknowns:
24 – 9 = 15
The expression on
the left-hand side...
...has the same value as the
expression on the right-hand side
Some equations contain unknown quantities, or variables.
2x – 3 = 5
The left-hand side...
...equals the right-hand side
The value of x that satisfies the equation is called the solution
(or root) of the equation.
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Topic
2.2.1
Properties of Equality
Addition and Subtraction in Equations
Addition Property of Equality
Subtraction Property of Equality
For any real numbers a, b, and c,
if a = b, then a + c = b + c.
For any real numbers a, b, and c,
if a = b, then a – c = b – c.
These properties mean that adding or subtracting the same number on
both sides of an equation will give you an equivalent equation.
This may allow you to isolate the variable on one side of the equals sign.
Finding the possible values of the variables in an equation is called
solving the equation.
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Topic
2.2.1
Example
Properties of Equality
1
Solve x + 9 = 16.
Solution
x + 9 = 16
You want x on its own, but here x has 9 added to it.
(x + 9) – 9 = 16 – 9
So subtract 9 from both sides to get x on its own.
x + (9 – 9) = 16 – 9
Now simplify the equation to find x.
x + 0 = 16 – 9
x = 16 – 9
x=7
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Solution follows…
Topic
2.2.1
Properties of Equality
In the last Example, we solved x + 9 = 16 and found that x = 7
is the root of the equation.
If x takes the value 7, then the equation is satisfied.
If x takes any other value, then the equation is not satisfied.
For example, if x = 6, then the left-hand side has the value
6 + 9 = 15, which does not equal the right-hand side, 16.
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Topic
2.2.1
Properties of Equality
When you’re actually solving equations, you won’t need to go
through all the stages each time — but it’s really important that
you understand the theory of the properties of equality.
• If you have a “+ 9” that you don’t want, you can get
rid of it by just subtracting 9 from both sides.
• If you have a “– 9” that you want to get rid of,
you can just add 9 to both sides.
In other words, you just need to use the inverse operations.
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Topic
2.2.1
Example
Properties of Equality
2
Solve x + 10 = 12.
Solution
x + 10 = 12
Given equation
x = 12 – 10
Subtract 10 from both sides
x=2
9
Solution follows…
Topic
2.2.1
Example
Properties of Equality
3
Solve x – 7 = 8.
Solution
x–7=8
Given equation
x=8+7
Add 7 to both sides
x = 15
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Solution follows…
Topic
2.2.1
Properties of Equality
Guided Practice
In Exercises 1–8, solve the equation for the unknown variable.
1. x + 7 = 15
x=8
4. x – (–9) = –17 x = –26
3.
5. –9 + x = 10
7.
2. x + 2 = –8 x = –10
x = 19
6. x – 0.9 = 3.7
x = 4.6
8. –0.5 = x – 0.125 x = –0.375
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Solution follows…
Topic
2.2.1
Properties of Equality
Multiplication and Division in Equations
Multiplication Property of Equality
Division Property of Equality
For any real numbers a, b, and c,
if a = b, then a × c = b × c.
For any real numbers a, b, and c,
such that c  0, if a = b, then
.
These properties mean that multiplying or dividing by the same number
on both sides of an equation will give you an equivalent equation.
This can help you to isolate the variable and solve the equation.
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Topic
2.2.1
Properties of Equality
Multiply or Divide to Get the Variable on Its Own
As with addition and subtraction, you can get the variable on
its own by simply performing the inverse operation.
• If you have “× 3” on one side of the equation, you
can get rid of that value by dividing both sides by 3.
• If you have a “÷ 3” that you want to get rid of,
you can just multiply both sides by 3.
Once again, you just need to use the inverse operations.
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Topic
2.2.1
Example
Properties of Equality
4
Solve 2x = 18.
Solution
2x = 18
You want x on its own... but here you’ve got 2x.
Divide both sides by 2 to get x on its own
Now simplify the equation to find x.
1x = 9
x=9
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Solution follows…
Topic
2.2.1
Example
Properties of Equality
5
Solve
Solution
Multiply both sides by 3 to get m on its own
m = 21
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Solution follows…
Topic
Properties of Equality
2.2.1
Example
6
Solve 4x – 2(2x – 1) = 2 – x + 3(x – 4).
Solution
Some equations are a bit more complicated. Take them step by step.
4x – 2(2x – 1) = 2 – x + 3(x – 4)
Given equation
4x – 4x + 2 = 2 – x + 3x – 12
Clear out any grouping symbols
2 = –10 + 2x
Then combine like terms
2x = 12
x=6
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Solution follows…
Topic
2.2.1
Properties of Equality
Guided Practice
In Exercises 9–16, solve each equation for the unknown variable.
9. 4x = 144
11.
10. –7x = –7
x = 36
x = –28
13. –3x + 4 = 19
x=1
12.
x = –5
14. 4 – 2x = 18
x = –7
15. 3x – 2(x – 1) = 2x – 3(x – 4)
16. 3x – 4(x – 1) = 2(x + 9) – 5x
x=5
x=7
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Solution follows…
Topic
2.2.1
Properties of Equality
Independent Practice
Solve each of these equations:
1. –2(3x – 5) + 3(x – 1) = –5
2. 4(2a + 1) – 5(a – 2) = 8
x=4
a = –2
3. 5(2x – 1) – 4(x – 2) = –15
x = –3
4. 2(5m + 7) – 3(3m + 2) = 4m
5. 4(5x + 2) – 5(3x + 1) = 2(x – 1)
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Solution follows…
Topic
2.2.1
Properties of Equality
Independent Practice
Solve each of these equations:
6. b – {3 – [b – (2 – b) + 4]} = –2(–b – 3)
b=7
7. 4[3x – 2(3x – 1) + 3(2x – 1)] = 2[–2x + 3(x – 1)] – (5x – 1)
8. 30 – 3(m + 7) = –3(2m + 27)
m = –30
9. 8x – 3(2x – 3) = –4(2 – x) + 3(x – 4) – 1
x=6
10. –5x – [4 – (3 – x)] = –(4x + 6)
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Solution follows…
Topic
2.2.1
Properties of Equality
Independent Practice
In Exercises 11–17, solve the equations and check your solutions.
You don’t need to show all your steps.
11. 4t = 60
t = 15
12. x + 21 = 19
13.
p=8
14. 7 – y = –11
15.
y = 28
16. 40 – x = 6
17.
x = –2
y = 18
x = 34
s = 16
18. Solve –(3m – 8) = 12 – m
m=–2
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Solution follows…
Topic
2.2.1
Properties of Equality
Independent Practice
19. Denzel takes a two-part math test. In the first part he gets 49 points
and in the second part he gets
of the x points. If his overall grade for
the test was 65, find the value of x.
total points =
; x = 36
20. Latoya takes 3 science tests. She scores 24%, 43%, and x% in
the tests. Write an expression for her average percentage over the
3 tests. Her average percentage is 52. Calculate the value of x.
; x = 89
Solve the following equations. Show all your steps and justify them
by citing the relevant properties.
21. x + 8 = 13 x = 5
22. 11 + y = 15 y = 4
23. y – 7 = 19
25. 4m = 16
y = 26
m=4
24.
x = 12
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Solution follows…
Topic
2.2.1
Properties of Equality
Round Up
Solving an equation means isolating the variable.
Anything you don’t want on one side of the equation can be
“taken over to the other side” by using the inverse operation.
You’re always aiming for an expression of the form:
“x = ...” (or “y = ...,” etc.).
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