Grade 7 Algebra-Expressions and Equations
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Answer the questions
(1) 16 years ago, the age of Phoebe's aunt was 4 times the age of Phoebe. If, her aunt is 76 years old now, how old is Phoebe now?
(2) In a collection of yellow, green, and black marbles, there are 15% more yellow marbles than green marbles, and there are 50% more black marbles than yellow marbles. If C is the number of yellow marbles, what is the total number of marbles?
(3) Find the sum of following polynomials:
A) 2p3 - 8p2q - 18pq + 2q + 6 and 16p3 + 14p + 2 B) 9y3 - xy2 - 10xy + 10y - 8 and -5x3 + 2x2y + 8x - 4
(4) By following the shown pattern, find how many bulbs will be there in "Pattern y".
Pattern 1 Pattern 2 Pattern 3
(5) If f(x) = x2, find the value of .
(6) If the number in the center is the sum of the rest of the numbers, find the value of a.
1 3
12
2a 2
(7) A rectangular ground's length is 3 times its breadth. If its perimeter is 48 meters, what is the length and the breadth of the ground?
(10) Find the value of y from the equation: -5y + 2
4y - 1 = -9 7 . (11) Simplify:
A) -6x2 + 8y2 - 3xy + 9 - [ 5x - 9y + 5 - { 6x2 + 6 } ]
B) q - 16 - [ 5p2 - 16q2 - 9p - 10q - { 14p2 - 16pq - p + 4q + 14 } ] (12) Find the greatest n digit number.
Choose correct answer(s) from the given choices
(13) If 9x + 9 = 54, what will be the value of 9x - 9?
a. 35 b. 36
c. 41 d. 45
(14) Asaad is standing in a row. If he is xth from the left and yth from the right, how many people are standing in the row?
a. x + y - 1 b. x + y
c. x + y + 1 d. None of these
Check True/False
(15) An equation with single variable, where lowest power is one, is called a linear equation.
True False
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Answers
(1) 31 years
Step 1
We know that the current age of Phoebe's aunt is 76 years. Therefore, her aunt's age, 16 years ago = 76 - 16
= 60 years.
Step 2
It is given that, 16 years ago Phoebe's age was 4 times her aunt's age. Therefore, Phoebe's age 16 years ago = (aunt's age) / 4 = 60 / 4
= 15 years.
Step 3
So, Phoebe's current age = 15 + 16 = 31 years.
(2) 155 46 C
Step 1
According to the question, the number of yellow marbles is C. Let the number of green and black marbles be X and Y, respectively.
Step 2
There are 15% more yellow marbles than green marbles. This means:
C = X + 15% of X C = X + 15X
100 100C = 100X + 15X X = 100C
115 Step 3
There are 50% more black marbles than yellow marbles. This means:
Y = C + 50% of C Y = C + 50C
100 Y = 100C + 50C
100 Y = 150C
100 Step 4
Total number of marbles = Number of yellow marbles + Number of green marbles + Number of black marbles
= C + X + Y
= C + 100C
115 + 150C 100
= 11500C + 10000C + 17250C
11500
= 38750C 11500
= 155 46 C Step 5
Therefore, the total number of marbles is 155 46 C.
(3) A) 18p3 - 8p2q - 18pq + 14p + 2q + 8
Step 1
In order to add the given polynomials, let us first place the like terms one below other and add as usual.
+2p3 -8p2q -18pq +2q +6
+16p3 +14p +2
+18p3 -8p2q -18pq +14p +2q +8
Step 2
Sum of the polynomials 2p3 - 8p2q - 18pq + 2q + 6 and 16p3 + 14p + 2 is 18p3 - 8p2q - 18pq + 14p + 2q + 8.
B) -5x3 + 9y3 + 2x2y - xy2 - 10xy + 8x + 10y - 12
Step 1
In order to add the given polynomials, let us first place the like terms one below other and add as usual.
+9y3 -xy2 -10xy +10y -8
-5x3 +2x2y +8x -4
-5x3 +9y3 +2x2y -xy2 -10xy +8x +10y -12
Step 2
Sum of the polynomials 9y3 - xy2 - 10xy + 10y - 8 and -5x3 + 2x2y + 8x - 4 is -5x3 + 9y3 + 2x2y - xy2 - 10xy + 8x + 10y - 12.
(4) y2 + 2y + 2
Step 1
If we look at the shown patterns carefully, we notice that the patterns are following the rule (y+2)y + 2, where y is the number of patterns.
Step 2
Therefore, the number of bulbs shown in Pattern y are y2 + 2y + 2.
(5) 2x + a
Step 1
According to the question, f(x) = x2. Or, f(x + a) = (x + a)2
Step 2
Now, = (x + a)2 - x2 a
= x2 + a2 + 2xa - x2 a
= a2 + 2xa a
= a(a + 2x) a
= a + 2x
= 2x + a Step 3
Therefore, the value of is 2x + a.
(6) 3
Step 1
It is given that 12 is the sum of 1, 3, 2a, and 2 Therefore, 1 + 3 + 2a + 2 = 12
⇒ 2a + 6 = 12
⇒ 2a = 12 - 6
⇒ a = 6 2
⇒ a = 3 Step 2
Thus, the value of a is 3.
(7) 18 meters, 6 meters
Step 1
It is given that the length of the rectangular ground is 3 times its breadth.
Let us assume that the breadth of the ground is w.
So, the length of the ground will be 3w.
Step 2
Since, we know that the perimeter of a rectangle is twice the sum of the length and the breadth of the rectangle.
Perimeter of the ground = 2(w + 3w) Step 3
It is given that the perimeter of the rectangular ground is 48 meters.
Therefore, 2(w + 3w) = 48
⇒ 2(4w) = 48
⇒ 8w) = 48
⇒ w = 48 8
⇒ w = 6 Step 4
Breadth of the rectangular ground = w = 6 meters. Thus, the length of the rectangular ground is 3w
= 3 × 6
= 18 meters
(8) 7x2 - 6x - 2
Step 1
Total number of eggs produced by the poultry farm in a year = 15x2 - 2 The number of eggs packed = 2x2 + 5x - 8
The number of eggs destroyed = 6x2 + x + 8 Step 2
Number of Eggs left with the poultry farm = Total number of eggs produced by the poultry farm - Number of eggs packed - Number of eggs destroyed
= 15x2 - 2 - (2x2 + 5x - 8) - (6x2 + x + 8)
= 15x2 - 2 + ( -2x2 - 5x + 8) + ( -6x2 - x - 8)
= 7x2 - 6x - 2
Therefore, the number of eggs left in the poultry farm are 7x2 - 6x - 2.
(9) 50, 33
Step 1
Let us assume that the first number is x.
Since, the sum of the two numbers is 83, the second number is 83 - x.
Step 2
It is also given that the difference of the two numbers is 17.
Therefore, x - (83 - x) = 17
⇒ x - 83 + x = 17
⇒ 2x - 83 = 17
⇒ 2x = 17 + 83
⇒ 2x = 100
⇒ x = 100 2
⇒ x = 50 Step 3
Thus, the first number = 50 The second number = 83 - x
= 83 - 50
= 33
(10) y = -5
Step 1 We have:
-5y + 2
4y - 1 = -9 7
⇒ ( -5y + 2) × 7 = (4y - 1) × -9
⇒ -35y + 14 = -36y + 9
⇒ -35y + 36y = 9 - 14
⇒ 1y = -5
⇒ y = -5 1
⇒ y = -5 Step 2
Therefore, the value of y is -5.
(11) A) 8y2 - 3xy - 5x + 9y + 10
Step 1
It is given that -6x2 + 8y2 - 3xy + 9 - [ 5x - 9y + 5 - { 6x2 + 6 } ] Step 2
First we will open the smaller bracket inside:
-6x2 + 8y2 - 3xy + 9 - [ 5x - 9y + 5 -6x2 - 6 ]
= -6x2 + 8y2 - 3xy + 9 - [ -6x2 + 5x - 9y - 1 ] Step 3
Now, we will open the bigger bracket:
-6x2 + 8y2 - 3xy + 9 + 6x2 - 5x + 9y + 1
= 8y2 - 3xy - 5x + 9y + 10
B) 9p2 + 16q2 - 16pq + 8p + 15q - 2
Step 1
It is given that q - 16 - [ 5p2 - 16q2 - 9p - 10q - { 14p2 - 16pq - p + 4q + 14 } ] Step 2
First we will open the smaller bracket inside:
q - 16 - [ 5p2 - 16q2 - 9p - 10q -14p2 + 16pq + p - 4q - 14 ]
= q - 16 - [ -9p2 - 16q2 + 16pq - 8p - 14q - 14 ] Step 3
Now, we will open the bigger bracket:
q - 16 + 9p2 + 16q2 - 16pq + 8p + 14q + 14
= 9p2 + 16q2 - 16pq + 8p + 15q - 2
(12) 10n-1
Step 1
Since, we know that the greatest 1 digit number is 9.
Or, we can say that the greatest 1 digit number = 101 - 1 = 10 - 1 = 9
= 10n-1 Step 2
Thus, the greatest n digit number = 10n-1
(13) b. 36
Step 1
If we look at the question carefully, we notice that we have to find out the value of 9 x - 9.
Step 2
Firstly, we have to find the value of x from the expression 9x + 9 = 54.
⇒ 9x = 54 - 9
⇒ x = 45 9
⇒ x = 5 Step 3
Let us now place the value of x as 5, in the expression 9x - 9.
9x - 9 = (9 × 5) - 9
= 45 - 9
= 36 Step 4
Thus, the correct answer is option b.
(14) a. x + y - 1
Step 1
Since, Asaad is xth from the front, there are (x - 1) people who are standing in front of him.
Step 2
Similarly, since he is yth from the back, there are (y - 1) people who are standing behind him.
Step 3
Total number of people in the row =
Number of people standing in front of Asaad + 1 (Asaad himself) + Number of people standing behind Asaad
= (x - 1) + 1 + (y - 1)
= x + y - 1 Step 4
Hence, option a is the correct answer.
(15) False
Step 1
A linear equation is a first degree equation. Alternatively, we can say that the highest power of a variable in a linear equation is one. A linear equation in one variable is an equation that can be written in the form: ax + b = c, where a, b, and c are real numbers and a ≠ 0.
Here is an example:
2x + 5 = 6.
Step 2
Therefore, the statement "An equation with single variable, where lowest power, is one is called linear equation" is false.