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Chapter 7: Algebraic Expressions Exercise – 7.4 Question: 1 Simplify, the algebraic expressions by removing grouping symbols. 2x + (5x – 3y) Solution: 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, the algebraic expressions by removing grouping symbols. 3x – (y – 2x) Solution: 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, the algebraic expressions by removing grouping symbols. 5a – (3b – 2a + 4c) Solution: 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, the algebraic expressions by removing grouping symbols. -2(x2 - y2 + xy) - 3(x2 +y2 - xy) Solution: 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. Therefore, we have = -2x2 + 2y2 - 2xy - 3x2 - 3y2 + 3xy = -2x2 - 3x2 + 2y2 - 3y2 - 2xy + 3xy = -5x2 - y2 + xy Question: 5 Simplify, the algebraic expressions by removing grouping symbols. 3x + 2y – {x – (2y – 3)} Solution: 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, the algebraic expressions by removing grouping symbols. 5a – {3a – (2 – a) + 4} Solution: 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, the algebraic expressions by removing grouping symbols. a – [b – {a – (b – 1) + 3a}] Solution: 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, the algebraic expressions by removing grouping symbols. a – [2b – {3a – (2b – 3c)}] Solution: 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, the algebraic expressions by removing grouping symbols. -x + [5y – {2x – (3y – 5x)}] Solution: 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, the algebraic expressions by removing grouping symbols. 2a – [4b – {4a – 3(2a – b)}] Solution: 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, the algebraic expressions by removing grouping symbols. -a – [a + {a + b – 2a – (a – 2b)} - b] Solution: 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, the algebraic expressions by removing grouping symbols. 2x – 3y – [3x – 2y -{x – z – (x – 2y)}] Solution: 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, the algebraic expressions by removing grouping symbols. 5 + [x – {2y – (6x + y – 4) + 2x} – {x – (y – 2)}] Solution: 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, the algebraic expressions by removing grouping symbols. x2 - [3x + [2x - (x2 - 1)] + 2] Solution: 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, the algebraic expressions by removing grouping symbols. 20 - [5xy + 3[x2 - (xy - y) - (x - y)]] Solution: 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, the algebraic expressions by removing grouping symbols. 85 – [12x – 7(8x – 3) – 2{10x – 5(2 – 4x)}] Solution: 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, the algebraic expressions by removing grouping symbols. xy[yz – zx – {yx – (3y – xz) – (xy – zy)}] Solution: 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
Simplify, the algebraic expressions by removing grouping symbols.
2x + (5x – 3y)
We have
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
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
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
-2(x2 - y2 + xy) - 3(x2 +y2 - xy)
- 2(x2 - y2 + xy) - 3(x2 +y2 - xy)
= -2x2 + 2y2 - 2xy - 3x2 - 3y2 + 3xy
= -2x2 - 3x2 + 2y2 - 3y2 - 2xy + 3xy
= -5x2 - y2 + xy
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
5a – {3a – (2 – a) + 4}
= 5a – {3a – 2 + a + 4}
= 5a – 3a + 2 – a – 4
= 5a – 4a – 2
= a – 2
a – [b – {a – (b – 1) + 3a}]
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 – {4a – b + 1}]
= a – [b – 4a + b – 1]
= a – [2b – 4a – 1]
= a – 2b + 4a + 1
= 5a – 2b + 1
a – [2b – {3a – (2b – 3c)}]
First we have to remove the small brackets, or parentheses, ( ), then the curly brackets, { }, and then the square brackets, [ ].
= a – [2b – {3a – 2b + 3c}]
= a – [2b – 3a + 2b – 3c]
= a – [4b – 3a – 3c]
= a – 4b + 3a + 3c
= 4a – 4b + 3c
-x + [5y – {2x – (3y – 5x)}]
First we have to remove the small brackets, or parentheses, ( ), then the curly brackets { }, and then the square brackets, [ ].
– 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
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
-a – [a + {a + b – 2a – (a – 2b)} - b]
– 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
2x – 3y – [3x – 2y -{x – z – (x – 2y)}]
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
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
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
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
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
xy[yz – zx – {yx – (3y – xz) – (xy – zy)}]
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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Chapter 7: Algebraic Expressions Exercise –...