# Simplify Algebraic Expressions Using like terms - Grade 6

### Solve the following questions (solutions at the bottom of the page)

1. What are the like terms and their coefficients in each of the following expressions?
1. 7 x - 5 x + 6 + 3x
2. 6 x + 11 x + 9 + x - 7
3. 3 x + 9 y + 9 x - y
4. (1/3) x + (2/3) x - 3

2. Simplify the following expressions.
1. 2 x + x
2. 12 x - 5 x + 11 - 4 x
3. 9 x + 11 x + 19 - x - 7
4. 4 x + 8 y + 10 x - 3 y
5. x + 5 + 3 y + 6 x - x - 2 y + 6
6. (1/2) x + (3/2) x + 2

3. Simplify the following expressions and identify the ones that have the same terms (equivalent expressions).
1. 3 x + 3 + 4 x + 2
2. 4 x + 2 a + 6 x + 6a
3. x + 2 + 3 x + 3 x + 3
4. x - x + 3 y
5. 7 a + 10 x + a
6. 4 x + 3 y - 4 x

4. Use the distributive property then simplify the following expressions.
1. 3 (x + 1) + 5 x - 4
2. 3 (x - 1) + 2 (x + 2) + 5 x
3. 4 (x + 2) + 2 (y + 2) + x + 3 y
4. 3 (x / 3 + 1) + 2 (x / 2 - 1) + 3

## Solutions to the Above Questions

1. Solution
Like terms have the same variable raised to the same power. Hence

1. in the expression 7 x - 5 x + 6 + 3 x     7 x , - 5 x and 3 x are like terms
The coefficients of the like terms 7 x , - 5 x and 3 x are 7 , - 5 and 3 respectively.

2. in the expression 6 x + 11 x + 9 + x - 7     6 x , 11 x and x are like terms; 9 and -7 are numbers and therefore like terms.
The coefficients of the like terms 6 x , 11 x and x are 6 , 11 and 1 respectively.
9 and -7 are numbers.

3. in the expression 3 x + 9 y + 9 x - y     there are two groups of like terms:    1) 3 x and 9 x are like terms     2) also 9 y and - y are like terms
The coefficients of the like terms 3 x and 9 x in group 1) are 3 and 9 respectively.
The coefficients of the like terms 9 y and - y in group 2) are 9 and - 1 respectively.

4. in (1 / 3) x + (2 / 3) x - 3     (1 / 3) x and (2 / 3) x are like terms and their coefficients are 1 / 3 and 2 / 3 respectively.

2. Solution
To simplify algebraic expressions, we first group and then add and/or subtract the coefficients of the like terms (see examples above).
1. 2 x + x     Given
= (2 + 1) x     identify coefficients and put variable out of parentheses (factoring)
= 3 x     add coefficients to simplify

2. 12 x - 5 x + 11 - 4 x     Given
= (12 x - 5 x - 4 x) + 11     use parentheses to put like terms together
= (12 - 5 - 4) x + 11     identify coefficients and put vaiable out of parentheses (factor x out)
= 3 x + 11     add and subtract coefficients to simplify

3. 9 x + 11 x + 19 - x - 7     Given
= (9 x + 11 x - x) + (19 - 7)     use parentheses to put like terms together
= (9 + 11 - 1) x + (19 - 7)     identify coefficients and put vaiable out of parentheses (factor x out)
= 19 x + 12     add and subtract coefficients to simplify

4. 4 x + 8 y + 10 x - 3 y     Given expression has two variables
= (4 x + 10 x) + (8 y - 3 y)     use parentheses to put like terms together
= (4 + 10) x + (8 - 3) y     identify coefficients and put variables x and y out of parentheses (factor x out)
= 14 x + 5 y     add and/or subtract coefficients to simplify

5. x + 5 + 3 y + 6 x - x - 2 y + 6     Given expression has two variables
= ( x + 6 x - x) + (3 y - 2 y ) + ( 5 + 6 )     use parentheses to put like terms together
= ( 1 + 6 - 1) x + (3 - 2 ) y + ( 5 + 6 )     identify coefficients and put variables x and y out of parentheses (factor x out)
= 6 x + y + 11     simplify

6. (1/2) x + (3/2) x + 2     Given expression has fractions
= ( (1/2) x + (3/2) x ) + 2     use parentheses to put like terms together
= ( 1 / 2 + 3 / 2) x + 2     identify coefficients and put variables x and y out of parentheses (factor x out)
= (3/3) x + 2
= x + 2     simplify

3. Solution
We first simplify the given expressions (see exercise 2 above)
1. 3 x + 3 + 4 x + 2 = (3 x + 4 x) + (3 + 2) = (3 + 4 ) x + (3 + 2) = 7 x + 5
2. 4 x + 2 a + 6 x + 6a = (4 x + 6 x) + (2 a + 6 a) = (4 + 6 ) x + (2 + 6 ) a = 10 x + 8 a
3. x + 2 + 3 x + 3 x + 3 = (x + 3 x + 3 x) + (2 + 3) = (1 + 3 + 3 ) x + (2 + 3) = 7 x + 5
4. x - x + 3 y = (1 - 1) x + 3 y = 0 x + 3 y = 0 + 3 y = 3 y
5. 7 a + 10 x + a = (7 a + a) + 10 x = (7 + 1) a + 10 x = 8 a + 10 x
6. 4 x + 3 y - 4 x = (4 x - 4 x) + 3 y = (4 - 4) x + 3 y = 0 x + 3 y = 0 + 3 y = 3 y

Conclusion
The expressions in parts a) and c) are equivalent;
the expressions in parts b) and e) are equivalent
and the expressions in parts d) and f) are equivalent.

4. Solution
We first use the distributive property to take out the parentheses and then simplify.

1. 3 (x + 1) + 5 x - 4     Given
= (3)(x) + (3)(1) + 5 x - 4     Use distributive property to multiply
= 3 x + 3 + 5 x - 4     Multiply and simplify
= (3 x + 5 x) + (3 - 4)     use parentheses to put like terms together
= (3 + 5) x + (3 - 4)     identify coefficients and put variables x out of parentheses (factoring)
= 8 x - 1     simplify

2. 3 (x - 1) + 2 (x + 2) + 5 x     Given
=(3)(x) + (3)(-1) + (2)(x) + (2)(2) + 5 x     Use distributive property to multiply
= 3 x - 3 + 2 x + 4 + 5 x     Multiply and simplify
= (3 x + 2 x + 5 x) + (- 3 + 4)     use parentheses to put like terms together
= (3 + 2 + 5) x + (-3 + 4)     identify coefficients and put variables x out of parentheses (factoring)
= 10 x + 1     simplify

3. 4 (x + 2) + 2 (y + 2) + x + 3 y     Given
= (4)(x) + (4)(2) + (2)(y) + (2)(2) + x + 3 y     Use distributive property to multiply
= 4 x + 8 + 2 y + 4 + x + 3 y     Multiply and simplify
= (4 x + x) + (2 y + 3 y) + (8 + 4)     use parentheses to put like terms together
= (4 + 1) x + (2 + 3) y + (8 + 4)     identify coefficients and put variables x out of parentheses (factoring)
= 5 x + 5 y + 12     simplify

4. 3 (x / 3 + 1) + 2 (x / 2 - 1) + 3     Given
= (3)(x / 3) + (3)(1) + (2)(x / 2) + (2)(-1) + 3     Use distributive property to multiply
= 3 x / 3 + 3 + 2 x / 2 - 2 + 3     Multiply and simplify
= ( 3 x / 3 + 3 + 2 x / 2 ) + (3 - 2 + 3)     use parentheses to put like terms together
= (3 / 3 + 2 / 2) x + (3 - 2 + 3)     identify coefficients and put variables x out of parentheses (factoring)
= (1 + 1) x + 4 = 2 x + 4     simplify

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