Exercise 7.4
Page-7.20Question 1:
Simplify each of the following algebraic expressions by removing grouping symbols.
2x + (5x − 3y)
Answer 1:
We have
2x + (5x − 3y)
Since the '+' sign precedes the parentheses, we have to retain the sign of each term in the parentheses when we remove them.
= 2x + 5x - 3y
= 7x - 3y
Question 2:
Simplify each of the following algebraic expressions by removing grouping symbols.
3x − (y − 2x)
Answer 2:
We have
3x − (y − 2x)
Since the '-' sign precedes the parentheses, we have to change the sign of each term in the parentheses when we remove them. Therefore, we have
3x − y + 2x
= 5x - y
Question 3:
Simplify each of the following algebraic expressions by removing grouping symbols.
5a − (3b − 2a + 4c)
Answer 3:
We have
5a − (3b − 2a + 4c)
Since the '-' sign precedes the parentheses, we have to change the sign of each term in the parentheses when we remove them.
= 5a - 3b + 2a - 4c
= 7a - 3b - 4c
Question 4:
Simplify each of the following algebraic expressions by removing grouping symbols.
−2 (x2 − y2 + xy) − 3(x2 + y2 − xy)
Answer 4:
We have
− 2(x2 − y2 + xy) − 3(x2 + y2 − xy)
Since the '-' sign precedes the parentheses, we have to change the sign of each term in the parentheses when we remove them.
= - 2x2 + 2y2 - 2xy - 3x2 - 3y2 + 3xy
= - 2x2 - 3x2 + 2y2- 3y2 - 2xy + 3xy
= - 5x2 - y2 + xy
Question 5:
Simplify each of the following algebraic expressions by removing grouping symbols.
3x + 2y − {x − (2y − 3)}
Answer 5:
We have
3x + 2y − {x − (2y − 3)}
First, we have to remove the small brackets (or parentheses): ( ). Then, we have to remove the curly brackets (or braces): { }.
Therefore,
= 3x + 2y − {x − 2y + 3}
= 3x + 2y − x + 2y - 3
= 2x + 4y - 3
Question 6:
Simplify each of the following algebraic expressions by removing grouping symbols.
5a − {3a − (2 − a) + 4}
Answer 6:
We have
5a − {3a − (2 − a) + 4}
First, we have to remove the small brackets (or parentheses): ( ). Then, we have to remove the curly brackets (or braces): { }.
Therefore,
= 5a − {3a − 2 + a + 4}
= 5a − 3a + 2 - a - 4
= 5a - 4a - 2
= a - 2
Question 7:
Simplify each of the following algebraic expressions by removing grouping symbols.
a − [b − {a − (b − 1) + 3a}]
Answer 7:
First we have to remove the parentheses, or small brackets, ( ), then the curly brackets, { }, and then the square brackets [ ].
Therefore, we have
a - [b - {a - (b - 1) + 3a}]
= a - [b - {a - b + 1 + 3a}]
= a - [b - {4a - b + 1}]
= a - [b - 4a + b - 1]
= a - [2b - 4a - 1]
= a - 2b + 4a + 1
= 5a - 2b + 1
Question 8:
Simplify each of the following algebraic expressions by removing grouping symbols.
a − [2b − {3a − (2b − 3c)}]
Answer 8:
First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
a - [2b - {3a - (2b - 3c)}]
= a - [2b - {3a - 2b + 3c}]
= a - [2b - 3a + 2b - 3c]
= a - [4b - 3a - 3c]
= a - 4b + 3a + 3c
= 4a - 4b + 3c
Question 9:
Simplify each of the following algebraic expressions by removing grouping symbols.
−x + [5y − {2x − (3y − 5x)}]
Answer 9:
First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
- x + [5y - {2x - (3y - 5x)}]
= - x + [5y - {2x - 3y + 5x}]
= - x + [5y - {7x - 3y}]
= - x + [5y - 7x + 3y]
= - x + [8y - 7x]
= - x + 8y - 7x
= - 8x + 8y
Question 10:
Simplify each of the following algebraic expressions by removing grouping symbols.
2a − [4b − {4a − 3(2a − b)}]
Answer 10:
First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
2a - [4b - {4a - 3(2a - b)}]
= 2a - [4b - {4a - 6a + 3b}]
= 2a - [4b - {- 2a + 3b}]
= 2a - [4b + 2a - 3b]
= 2a - [b + 2a]
= 2a - b - 2a
= - b
Question 11:
Simplify each of the following algebraic expressions by removing grouping symbols.
−a − [a + {a + b − 2a − (a − 2b)} − b]
Answer 11:
First we have to remove the small brackets, or parentheses, ( ), then the curly brackets,{ }, and then the square brackets, [ ].
Therefore, we have
- a - [a + {a + b - 2a - (a - 2b)} - b]
= - a - [a + {a + b - 2a - a + 2b} - b]
= - a - [a + {- 2a + 3b} - b]
= - a - [a - 2a + 3b - b]
= - a - [- a + 2b]
= - a + a - 2b
= - 2b
Question 12:
Simplify each of the following algebraic expressions by removing grouping symbols.
2x − 3y − [3x − 2y − {x − z − (x − 2y)}]
Answer 12:
First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
2x - 3y - [3x - 2y - {x - z - (x - 2y)}]
= 2x - 3y - [3x - 2y - {x - z - x + 2y}]
= 2x - 3y - [3x - 2y - {- z + 2y}]
= 2x - 3y - [3x - 2y + z - 2y]
= 2x - 3y - [3x - 4y + z]
= 2x - 3y - 3x + 4y - z
= - x + y - z
Question 13:
Simplify each of the following algebraic expressions by removing grouping symbols.
5 + [x − {2y − (6x + y − 4) + 2x} − {x − (y − 2)}]
Answer 13:
First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
5 + [x - {2y - (6x + y - 4) + 2x} - {x - (y - 2)}]
= 5 + [x - {2y - 6x - y + 4 + 2x} - {x - y + 2}]
= 5 + [x - {y - 4x + 4} - {x - y + 2}]
= 5 + [x - y + 4x - 4 - x + y - 2]
= 5 + [4x - 6]
= 5 + 4x - 6
= 4x - 1
Question 14:
Simplify each of the following algebraic expressions by removing grouping symbols.
x2 − [3x + {2x − (x2 − 1) + 2}]
Answer 14:
First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
x2 - [3x + {2x - (x2 - 1)} + 2]
= x2 - [3x + {2x - x2 + 1} + 2]
= x2 - [3x + 2x - x2 + 1+ 2]
= x2 - [5x - x2 + 3]
= x2 - 5x + x2 - 3
= 2x2 - 5x - 3
Question 15:
Simplify each of the following algebraic expressions by removing grouping symbols.
20 − [5xy + 3{x2 − (xy − y) − (x − y)}]
Answer 15:
First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
20 - [5xy + 3{x2 - (xy - y) - (x - y)}]
= 20 - [5xy + 3{x2 - xy + y - x + y}]
= 20 - [5xy + 3{x2 - xy + 2y - x}]
= 20 - [5xy + 3x2 - 3xy + 6y - 3x]
= 20 - [2xy + 3x2 + 6y - 3x]
= 20 - 2xy - 3x2 - 6y + 3x
= - 3x2 - 2xy - 6y + 3x + 20
Question 16:
Simplify each of the following algebraic expressions by removing grouping symbols.
85 − [12x − 7(8x − 3) − 2 {10x − 5(2 − 4x)}]
Answer 16:
First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
85 - [12x - 7(8x - 3) - 2{10x - 5(2 - 4x)}]
= 85 - [12x - 56x + 21 - 2{10x - 10 + 20x}]
= 85 - [12x - 56x + 21 - 2{30x - 10}]
= 85 - [12x - 56x + 21 - 60x + 20]
= 85 - [12x - 116x + 41]
= 85 - [- 104x + 41]
= 85 + 104x - 41
= 44 + 104x
Question 17:
Simplify each of the following algebraic expressions by removing grouping symbols.
xy [yz − zx − {yx − (3y − xz) − (xy − zy)}]
Answer 17:
First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
Therefore, we have
xy - [yz - zx - {yx - (3y - xz) - (xy - zy)}]
= xy - [yz - zx - {yx - 3y + xz - xy + zy}]
= xy - [yz - zx - {- 3y + xz + zy}]
= xy - [yz - zx + 3y - xz - zy]
= xy - [- zx + 3y - xz]
= xy - [- 2zx + 3y]
= xy + 2xz - 3y
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