Upload
darryl-chatterton
View
218
Download
1
Tags:
Embed Size (px)
Citation preview
1
Topic 2.2.1Topic 2.2.1
Properties of EqualityProperties of Equality
2
Topic2.2.1
Properties of EqualityProperties of Equality
California Standard:4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x – 5) + 4(x – 2) = 12.
What it means for you:You’ll solve one-step equations using properties of equality.
Key words:• expression• equation• variable• solve• isolate• equality
3
Topic2.2.1
Properties of EqualityProperties of Equality
Now it’s time to use the material on expressions you learned in Section 2.1.
In this Topic you’ll solve equations that involve addition, subtraction, multiplication, and division.
An equation contains two expressions, with an equals sign in the middle to show that they’re equal.
For example: 2x – 3 = 4x + 5
4
Topic2.2.1
An Equation Shows That Two Expressions are Equal
Properties of EqualityProperties of Equality
An equation is a way of stating that two expressions have the same value.
The expression on the left-hand side...
24 – 9 = 15...has the same value as the expression on the right-hand side
Some equations contain unknown quantities, or variables.
The left-hand side...
2x – 3 = 5...equals the right-hand side
The value of x that satisfies the equation is called the solution (or root) of the equation.
This equation contains only numbers — there are no unknowns:
5
Topic2.2.1
Addition and Subtraction in Equations
Properties of EqualityProperties of Equality
Subtraction Property of Equality
For any real numbers a, b, and c, if a = b, then a – c = b – c.
Finding the possible values of the variables in an equation is called solving the equation.
Addition Property of Equality
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.
6
Topic2.2.1
Example 1
Solution follows…
Properties of EqualityProperties of Equality
Solve x + 9 = 16.
Solution
x = 7
So subtract 9 from both sides to get x on its own.(x + 9) – 9 = 16 – 9
x + 0 = 16 – 9
x = 16 – 9
You want x on its own, but here x has 9 added to it.x + 9 = 16
x + (9 – 9) = 16 – 9 Now simplify the equation to find x.
7
Topic2.2.1
Properties of EqualityProperties 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.
8
Topic2.2.1
Properties of EqualityProperties of Equality
In other words, you just need to use the inverse operations.
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.
9
Topic2.2.1
Example 2
Solution follows…
Properties of EqualityProperties of Equality
Solve x + 10 = 12.
Solution
x + 10 = 12
x = 2
x = 12 – 10 Subtract 10 from both sides
Given equation
10
Topic2.2.1
Example 3
Solution follows…
Properties of EqualityProperties of Equality
Solve x – 7 = 8.
Solution
x – 7 = 8
x = 15
x = 8 + 7 Add 7 to both sides
Given equation
11
Topic2.2.1
Guided Practice
Solution follows…
Properties of EqualityProperties of Equality
1. x + 7 = 15 2. x + 2 = –8
3. 4. x – (–9) = –17
5. –9 + x = 10 6. x – 0.9 = 3.7
7. 8. –0.5 = x – 0.125
In Exercises 1–8, solve the equation for the unknown variable.
x = 8
x = 19
x = –10
x = –26
x = 4.6
x = –0.375
12
Topic2.2.1
Multiplication and Division in Equations
Properties of EqualityProperties of Equality
Multiplication Property of Equality
For any real numbers a, b, and c, if a = b, then a × c = b × c.
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.
Division Property of Equality
For any real numbers a, b, and c,
such that c 0, if a = b, then .
13
Topic2.2.1
Multiply or Divide to Get the Variable on Its Own
Properties of EqualityProperties of Equality
Once again, you just need to use the inverse operations.
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.
14
Topic2.2.1
Example 4
Solution follows…
Solve 2x = 18.
Solution
You want x on its own... but here you’ve got 2x.
x = 9
Divide both sides by 2 to get x on its own
1x = 9
2x = 18
Now simplify the equation to find x.
Properties of EqualityProperties of Equality
15
Topic2.2.1
Example 5
Solution follows…
Solve
Solution
m = 21
Multiply both sides by 3 to get m on its own
Properties of EqualityProperties of Equality
16
Topic2.2.1
Example 6
Solution follows…
Solve 4x – 2(2x – 1) = 2 – x + 3(x – 4).
Solution
4x – 2(2x – 1) = 2 – x + 3(x – 4)
Some equations are a bit more complicated. Take them step by step.
4x – 4x + 2 = 2 – x + 3x – 12
2 = –10 + 2x
x = 6
2x = 12
Clear out any grouping symbols
Then combine like terms
Given equation
Properties of EqualityProperties of Equality
17
Topic2.2.1
Guided Practice
Solution follows…
Properties of EqualityProperties of Equality
In Exercises 9–16, solve each equation for the unknown variable.
9. 4x = 144 10. –7x = –7
13. –3x + 4 = 19 14. 4 – 2x = 18
15. 3x – 2(x – 1) = 2x – 3(x – 4) 16. 3x – 4(x – 1) = 2(x + 9) – 5x
11. 12.
x = 36 x = 1
x = –28
x = –5 x = –7
x = 5 x = 7
18
Topic2.2.1
Independent Practice
Solution follows…
Properties of EqualityProperties of Equality
Solve each of these equations:
1. –2(3x – 5) + 3(x – 1) = –5 x = 4
2. 4(2a + 1) – 5(a – 2) = 8 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)
19
Topic2.2.1
Independent Practice
Solution follows…
Properties of EqualityProperties of Equality
6. b – {3 – [b – (2 – b) + 4]} = –2(–b – 3)
7. 4[3x – 2(3x – 1) + 3(2x – 1)] = 2[–2x + 3(x – 1)] – (5x – 1)
8. 30 – 3(m + 7) = –3(2m + 27)
9. 8x – 3(2x – 3) = –4(2 – x) + 3(x – 4) – 1
10. –5x – [4 – (3 – x)] = –(4x + 6)
b = 7
m = –30
x = 6
Solve each of these equations:
20
Topic2.2.1
Independent Practice
Solution follows…
Properties of EqualityProperties of Equality
In Exercises 11–17, solve the equations and check your solutions. You don’t need to show all your steps.
11. 4t = 60 12. x + 21 = 19
13. 14. 7 – y = –11
15. 16. 40 – x = 6
17.
t = 15 x = –2
p = 8 y = 18
y = 28 x = 34
s = 16
18. Solve –(3m – 8) = 12 – m m = – 2
21
Topic2.2.1
Independent Practice
Solution follows…
Properties of EqualityProperties of Equality
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.
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.
Solve the following equations. Show all your steps and justify them by citing the relevant properties.21. x + 8 = 13 22. 11 + y = 15
23. y – 7 = 19 24. 25. 4m = 16
total points =
; x = 89
x = 5 y = 4
y = 26 x = 12
m = 4
; x = 36
22
Topic2.2.1
Round UpRound Up
Properties of EqualityProperties of Equality
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.).