Algebra I/Content/Introduction to Basic Algebra Ideas/Solving Equations Using Properties of Mathematics
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Vocabulary
- Associative Property of Addition
- Associative Property of Multiplication
- Commutative Property of Addition
- Addition Property of Equality
- Subtraction Property of Equality
Lesson
Using many math rules and directions together. It is always a must to show the answer in its most broken down form for example if x=5/10 instead write x=1/2. It is the same but more simplified. The simplest answer is normally the objective.
Step back to previous lessons in this section before attempting this section.
Some math can be used to find a good golf swing among other things.
Associative Properties state that no matter how we group similar objects, the result will always be the same.
Commutative Properties state that no matter how we order similar objects, the result will always be the same.
Equality Properties state that if two numbers on either side of the equation are equal, and the operation processed is the same, as well having the same variables in the operations, the result will be the same for both sides of the equation. Read the section devoted to the Addition and Subtraction Properties of Equality for more information.
Associative Property of Addition
The associative property of addition shows us that when adding multiple values together, the outcome will always be the same. You can group numbers together in parenthesis, and it will still end up having the same outcome. For example, 3 + (4 + 5) = (3 + 4) + 5. The order remains the same, but the grouping has changed. The result, however, is consistent.
Associative Property of Multiplication
The associative properties work for both addition and multiplication. Think of grouping 3 * 3 together. You end up with 9. What about 3 * (3 * 4)? If we change the grouping here, the result will be the same. Try to visualize why this is. When multiplying, you're often building objects in rows and columns.
For example, a 2 inch * 4 inch block will be 2 inches across, and 4 downward. If you had a block 4 inches across and 2 inches downward, it would be the same size overall. When you put objects in parenthesis, remember to do those operations first. You may end up with certain "sides" of the operation or object being larger or smaller, but the total area will always have the same outcome.
Commutative Property of Addition
The commutative property of addition shows that no matter what order numbers are in when we add them, the result will always be the same. For example, (x + y) = (y + x), just how (3 + 4) = (4 + 3). Even though the order in which we added has shifted, the result doesn't change, and the statements on both sides of the equal sign remain true.
Addition Property of Equality
The addition property of equality states that if two variables or numbers are equal to each other on each side of the equation, and the operation they go through is alike, the resulting sum will be the same. If for example, both X and Y are 6, and you add Z to each of them: X = Y, (X + Z) = (Y + Z). 6 = 6, (6 + Z) = (6 + Z).
Subtraction Property of Equality
The subtraction property of equality states that if two variables or numbers are equal to each other on each side of the equation, and the operation they go through is alike, the resulting difference will be the same. If for example, both X and Y are 6, and you subtract Z from each of them: X = Y, (X - Z) = (Y - Z). 6 = 6, (6 - Z) = (6 - Z).
Example Problems
For examples 1,2 and 3 find x where y=6
Example 1:
(x+y)/2=14
substituting 6 for y Using the given y
(x+6)/2=14
By multiplying the denominator on either side simplifies it
2×((x+6)/2)=14×2
x+6=28
Minus 6 on either side does not affect the property of the given formula but again simplifies it
(x+6)-6=28-6
To arrive at the answer
Answer: x=22
Example 2:
(x-3y)+2x=15
again taking the hints given y=6
(x-3×6)+2x=15
Dealing with the parentheses
(x-18)+2x=15
taking down the parentheses is the most natural next move using P.E.M/D.A/S.
x-18+2x=15
adding x+2x
3x-18=15
adding 18 to either side
(3x-18)+18=15+18
taking down the parentheses
3x-18+18=33
naturally
3x=33
the property of the equation has not been changed but shifted like a tetris game to a desirable result one last step
divide the multiplier to either side to emerge the solution
3x/3=33/3
naturally
Answer: x=11
Example 3:
(15-y)/x=y²
sub y
(15-6)/x=6²
Parentheses
9/x=36
denominator
9=36x
divide the multiplier on either side
x=9/36
broken down further
Answer: x=1/4 or 0.25
Practice Games
Practice Problems
(Note: Answers are in parentheses after each problem)
Find x where y=9
correct to two decimals
This means round up or down accordingly until the answer is correct to two decimals
1) x=8(y/3) answer(x=24)
2) (x-4)=8+y answer(x=21)
3) (14+x)/y=3 answer(x=13)
4) (15-y)/x=y² answer(x=0.07)
5) (1+y²)/(x-4)=17-y³ answer(4.12)
6) x+7-(y-9)=43y+3x-y² answer(-149.5)
7) (12-x(45-y))/(y²-67)=84 answer(-32.33)