Algebra part three
Simplifying expressions in algebra
Before we can simplify expression using algebra, there are some rules (yeah, I know) that we must follow. Some of these rules may seem unnecessary, but they are a standard so that people all over the world can communicate in the same language - sort of. Maths is the language of symbols and text. (Some say it is the glue that holds the universe together!)
for multiplication of expressions
1) multiply all the numbers
2) Variables (letters) are written in alphabetical order
3) leave numbers in front (numbers are leaders, variables are followers)
Examples: 5 x 10x = 50x
3b x 4a = 12ab
2m x 4p x 3d=24dmp
Activity 1
Simplify the following expressions: x means times and x is a variable - just in case you get confused.
1) 5 x 3x
2) 8 x x
3) r x t
4) 7p x q
5) 4m x 6j x 2a
Answers: 15x, 8x, rt, 7pq, 48ajm
Simplify the following expressions: x means times and x is a variable - just in case you get confused.
1) 5 x 3x
2) 8 x x
3) r x t
4) 7p x q
5) 4m x 6j x 2a
Answers: 15x, 8x, rt, 7pq, 48ajm
addition and subtraction of expressions
Examples:
a) m + m + m + m = 4m
b) 10x - 2x = 8x
Activity 2
Simplify the following expressions
1) k + k + k
2) q + q + q + q + q
3) 3a + 5a
4) 4y + y
5) 16b - 8b
Answers: 3k, 5q, 8a, 5y, 8b
In the next section work left to right.
6) 7a + 5a - 3a
7) 50b - 10b + 2b
8) 4x - 2x - x
Answers: 12a - 3a=9a, 40b+2b=42b, 2x-x=x
a) m + m + m + m = 4m
b) 10x - 2x = 8x
Activity 2
Simplify the following expressions
1) k + k + k
2) q + q + q + q + q
3) 3a + 5a
4) 4y + y
5) 16b - 8b
Answers: 3k, 5q, 8a, 5y, 8b
In the next section work left to right.
6) 7a + 5a - 3a
7) 50b - 10b + 2b
8) 4x - 2x - x
Answers: 12a - 3a=9a, 40b+2b=42b, 2x-x=x
like and unlike terms
Like terms have the same letter or group of letters. For example, x and 2x are like terms, 3y and 7y are like terms.
xy, 3xy and -20xy are also like terms because they all contain the term xy. So if the terms are alike they can be added and subtracted together.
Examples of unlike terms are 8 and k or 5x and 5y. These terms cannot be added or subtracted together.
For each set of numbers write out the like terms.
Activity 3
1) {4a, 10a, 5k, a}
2) {3x, 11y, 21x, 2y}...more than one set of like terms here.
3) {4rt, 60kz, 38rt, 13pq}
4) does the set {4ab, 99ba} contain like terms?
Answers: {4a, 10a, a}, {3x, 21x} and {2y , 11y}, {4rt, 38rt}, yes because ab and ba are the same
xy, 3xy and -20xy are also like terms because they all contain the term xy. So if the terms are alike they can be added and subtracted together.
Examples of unlike terms are 8 and k or 5x and 5y. These terms cannot be added or subtracted together.
For each set of numbers write out the like terms.
Activity 3
1) {4a, 10a, 5k, a}
2) {3x, 11y, 21x, 2y}...more than one set of like terms here.
3) {4rt, 60kz, 38rt, 13pq}
4) does the set {4ab, 99ba} contain like terms?
Answers: {4a, 10a, a}, {3x, 21x} and {2y , 11y}, {4rt, 38rt}, yes because ab and ba are the same
Adding and subtracting like terms
Like terms can be grouped then added or subtracted together. For example,
Simplify 5x + 3 + 2x + 7. Using the associative property we can group (associate) the like terms together. We can then write:
5x + 3 + 2x + 7 = (5x + 2x) + (3 + 7)....x's are like terms and the numbers (constants) are like terms.
= 7x + 10 ...7x and 10 are not like terms so they can't be added together.
Simplify by collecting like terms and adding.
Activity 4
1) 4x + 6y + 2x + y 6) 4x + 3y - 2x + y
2) 7m + 2n + 3m + 8n 7) 11x + 7 - 4x + 1
3) p + q + 9p + 2q 8) 8x + 8 - x - 1
4) 5x + 3x + 5 + x 9) 16y + 2p - 2p - 15y...tricky!
5) k + k + k + j + j 10) 2x + 3 - 4x - 4...(negative numbers!)
Answers: 6x+7y, 10m+10n, 10p+3q, 9x+5, 3k+2j, 2x+4y, 7x+8, 7x+7, y since the p terms cancel out so 16y - 15y =y, -2x-1
Simplify 5x + 3 + 2x + 7. Using the associative property we can group (associate) the like terms together. We can then write:
5x + 3 + 2x + 7 = (5x + 2x) + (3 + 7)....x's are like terms and the numbers (constants) are like terms.
= 7x + 10 ...7x and 10 are not like terms so they can't be added together.
Simplify by collecting like terms and adding.
Activity 4
1) 4x + 6y + 2x + y 6) 4x + 3y - 2x + y
2) 7m + 2n + 3m + 8n 7) 11x + 7 - 4x + 1
3) p + q + 9p + 2q 8) 8x + 8 - x - 1
4) 5x + 3x + 5 + x 9) 16y + 2p - 2p - 15y...tricky!
5) k + k + k + j + j 10) 2x + 3 - 4x - 4...(negative numbers!)
Answers: 6x+7y, 10m+10n, 10p+3q, 9x+5, 3k+2j, 2x+4y, 7x+8, 7x+7, y since the p terms cancel out so 16y - 15y =y, -2x-1
Order of operations (bEDMAS)
The rules of BEDMAS also apply to algebra. Brackets and Exponents first (if there are any of course). Multiplication and division come before addition and subtraction. This topic was covered in Substitution.