CBSE Class 7 Mathematics Algebraic Expressions Assignment Set 05

Question 1. Add the following expressions:
(i) \( 5x^2y \); \( -7x^2y \); \( 9x^2y \)
(ii) \( \frac{3}{2}xy^2 \); \( -\frac{5}{7}xy^2 \); \( \frac{4}{15}xy^2 \)
(iii) \( 3xy^2 - 5x^2y \); \( -5xy^2 + 7x^2y \)
(iv) \( 2x^2 - 3y^2 \); \( 5x^2 - 2y^2 \); \( -x^2 + y^2 \)
Answer:
(i) Grouping the like terms together:
\( 5x^2y + (-7x^2y) + 9x^2y = (5 - 7 + 9)x^2y = 7x^2y \)

(ii) Combining the coefficients of \( xy^2 \):
\( \frac{3}{2}xy^2 + \left(-\frac{5}{7}xy^2\right) + \frac{4}{15}xy^2 = \left(\frac{3}{2} - \frac{5}{7} + \frac{4}{15}\right)xy^2 \)
Taking the LCM of denominators 2, 7, and 15, which is 210:
\( = \left(\frac{3 \times 105 - 5 \times 30 + 4 \times 14}{210}\right)xy^2 \)
\( = \left(\frac{315 - 150 + 56}{210}\right)xy^2 \)
\( = \frac{221}{210}xy^2 \)

(iii) Adding and combining like terms:
\( (3xy^2 - 5x^2y) + (-5xy^2 + 7x^2y) \)
\( = (3xy^2 - 5xy^2) + (-5x^2y + 7x^2y) \)
\( = -2xy^2 + 2x^2y \)

(iv) Collecting like terms:
\( (2x^2 - 3y^2) + (5x^2 - 2y^2) + (-x^2 + y^2) \)
\( = (2 + 5 - 1)x^2 + (-3 - 2 + 1)y^2 \)
\( = 6x^2 - 4y^2 \)
In simple words: Put terms with the same letters together and add their numbers.
Exam Tip: Check variable powers carefully so you only combine terms that match completely, such as \( xy^2 \) with \( xy^2 \).

Question 2. Subtract
(i) \( -8xy \) from \( 15xy \)
(ii) \( x^2 \) from \( -5x^2 \)
(iii) \( (x^2 - y^2) \) from \( (2x^2 - 3y^2 + 7) \)
(iv) \( \frac{2}{3}xy - \frac{5}{6}yz \) from \( -\frac{2}{3}yz + \frac{5}{6}xy \)
Answer:
(i) Writing the subtraction expression:
\( 15xy - (-8xy) = 15xy + 8xy = 23xy \)

(ii) Subtracting the given term:
\( -5x^2 - x^2 = (-5 - 1)x^2 = -6x^2 \)

(iii) Subtracting the first expression from the second:
\( (2x^2 - 3y^2 + 7) - (x^2 - y^2) \)
\( = 2x^2 - 3y^2 + 7 - x^2 + y^2 \)
\( = (2 - 1)x^2 + (-3 + 1)y^2 + 7 \)
\( = x^2 - 2y^2 + 7 \)

(iv) Setting up the difference:
\( \left(-\frac{2}{3}yz + \frac{5}{6}xy\right) - \left(\frac{2}{3}xy - \frac{5}{6}yz\right) \)
\( = -\frac{2}{3}yz + \frac{5}{6}xy - \frac{2}{3}xy + \frac{5}{6}yz \)
Group the like terms:
\( = \left(\frac{5}{6} - \frac{2}{3}\right)xy + \left(-\frac{2}{3} + \frac{5}{6}\right)yz \)
\( = \left(\frac{5 - 4}{6}\right)xy + \left(\frac{-4 + 5}{6}\right)yz \)
\( = \frac{1}{6}xy + \frac{1}{6}yz \)
In simple words: When you subtract, change the signs of the terms being taken away, then combine like terms.

Exam Tip: Remember that "subtract A from B" always means \( B - A \), not \( A - B \).

Question 3. Subtract \( (2a - 3b + 4c) \) from the sum of \( (a + b + c) \) and \( (-2a - 3b + 4c) \)
Answer:
First, find the sum of \( (a + b + c) \) and \( (-2a - 3b + 4c) \):
\( \text{Sum} = (a + b + c) + (-2a - 3b + 4c) \)
\( = (1 - 2)a + (1 - 3)b + (1 + 4)c \)
\( = -a - 2b + 5c \)

Next, subtract \( (2a - 3b + 4c) \) from this sum:
\( (-a - 2b + 5c) - (2a - 3b + 4c) \)
\( = -a - 2b + 5c - 2a + 3b - 4c \)
\( = (-1 - 2)a + (-2 + 3)b + (5 - 4)c \)
\( = -3a + b + c \)
In simple words: Add the first two expressions together, then subtract the other one from your total.

Exam Tip: Work out the sum first as a clear separate step before setting up the subtraction.

Question 4. Add \( (5x^2 - 7y^2 + 4z^2) \) to the sum of \( (7x^2 - 5y^2 + 3z^2) \) and \( (x^2 - y^2) \)
Answer:
First, find the sum of \( (7x^2 - 5y^2 + 3z^2) \) and \( (x^2 - y^2) \):
\( \text{Sum} = (7x^2 - 5y^2 + 3z^2) + (x^2 - y^2) \)
\( = (7 + 1)x^2 + (-5 - 1)y^2 + 3z^2 \)
\( = 8x^2 - 6y^2 + 3z^2 \)

Now, add \( (5x^2 - 7y^2 + 4z^2) \) to this result:
\( (8x^2 - 6y^2 + 3z^2) + (5x^2 - 7y^2 + 4z^2) \)
\( = (8 + 5)x^2 + (-6 - 7)y^2 + (3 + 4)z^2 \)
\( = 13x^2 - 13y^2 + 7z^2 \)
In simple words: Find the sum of the last two expressions, then add the first expression to it.

Exam Tip: Keep terms aligned by their variables and powers to avoid mixing up coefficients.

Question 5. Evaluate \( (5x - 9y) - (9x - 5y) \)
Answer:
Remove the brackets by changing the sign of each term inside the second bracket:
\( (5x - 9y) - (9x - 5y) \)
\( = 5x - 9y - 9x + 5y \)
Group like terms together:
\( = (5 - 9)x + (-9 + 5)y \)
\( = -4x - 4y \)
In simple words: Open the brackets, change the signs after the minus sign, and join like terms.

Exam Tip: Be careful with the double minus: \( -(-5y) \) turns into \( +5y \).

Question 6. Subtract the sum of \( (6x + 7y - 3z) \) and \( (3x - 4y + 5z) \) from the sum of \( (9x - 3y + 4z) \) and \( (x - y - z) \)
Answer:
First, find the sum of the first two expressions:
\( \text{First sum} = (6x + 7y - 3z) + (3x - 4y + 5z) \)
\( = (6 + 3)x + (7 - 4)y + (-3 + 5)z \)
\( = 9x + 3y + 2z \)

Next, find the sum of the second two expressions:
\( \text{Second sum} = (9x - 3y + 4z) + (x - y - z) \)
\( = (9 + 1)x + (-3 - 1)y + (4 - 1)z \)
\( = 10x - 4y + 3z \)

Now, subtract the first sum from the second sum:
\( (10x - 4y + 3z) - (9x + 3y + 2z) \)
\( = 10x - 4y + 3z - 9x - 3y - 2z \)
\( = (10 - 9)x + (-4 - 3)y + (3 - 2)z \)
\( = x - 7y + z \)
In simple words: Add each pair of expressions separately, then take the first sum away from the second sum.

Exam Tip: Clearly label "First Sum" and "Second Sum" in your steps to prevent subtracting in the reverse order.

Question 7. Subtract the sum of \( (8x - 6x^2 + 10) \) and \( (-10x - 8 + 8x^2) \) from \( (-7) \)
Answer:
First, find the sum of \( (8x - 6x^2 + 10) \) and \( (-10x - 8 + 8x^2) \):
\( \text{Sum} = (8x - 6x^2 + 10) + (-10x - 8 + 8x^2) \)
\( = (-6 + 8)x^2 + (8 - 10)x + (10 - 8) \)
\( = 2x^2 - 2x + 2 \)

Now, subtract this sum from \( -7 \):
\( -7 - (2x^2 - 2x + 2) \)
\( = -7 - 2x^2 + 2x - 2 \)
\( = -2x^2 + 2x - 9 \)
In simple words: Add the two expressions first, then subtract that total from -7.

Exam Tip: Distribute the negative sign across all three terms of the sum when opening the bracket.

Question 8. Simplify
(i) \( (x^2 - x) - \frac{1}{2}(x - 5x^2 + 3) \)
(ii) \( (\frac{4}{5}x^2 - \frac{1}{2}y^2 + 7) - \frac{3}{4}(\frac{4}{3}x^2 - \frac{8}{3}y^2 + 4) \)
Answer:
(i) Expand the expression by multiplying each term inside the bracket by \( -\frac{1}{2} \):
\( (x^2 - x) - \frac{1}{2}(x - 5x^2 + 3) \)
\( = x^2 - x - \frac{1}{2}x + \frac{5}{2}x^2 - \frac{3}{2} \)
Combine the like terms:
\( = \left(1 + \frac{5}{2}\right)x^2 + \left(-1 - \frac{1}{2}\right)x - \frac{3}{2} \)
\( = \frac{7}{2}x^2 - \frac{3}{2}x - \frac{3}{2} \)

(ii) Expand the second bracket by multiplying through by \( -\frac{3}{4} \):
\( -\frac{3}{4} \times \frac{4}{3}x^2 = -x^2 \)
\( -\frac{3}{4} \times \left(-\frac{8}{3}y^2\right) = +2y^2 \)
\( -\frac{3}{4} \times 4 = -3 \)

Now, rewrite and group like terms:
\( \left(\frac{4}{5}x^2 - \frac{1}{2}y^2 + 7\right) - (x^2 - 2y^2 + 3) \)
\( = \frac{4}{5}x^2 - \frac{1}{2}y^2 + 7 - x^2 + 2y^2 - 3 \)
\( = \left(\frac{4}{5} - 1\right)x^2 + \left(-\frac{1}{2} + 2\right)y^2 + (7 - 3) \)
\( = -\frac{1}{5}x^2 + \frac{3}{2}y^2 + 4 \)
In simple words: Multiply out the fractions outside the brackets, then group and simplify matching terms.

Exam Tip: Cancel common factors before multiplying fractions to make the simplification much easier.

Question 9. Simplify
(i) \( (\frac{1}{3}x^2 - \frac{4}{7}x + 5) - (\frac{2}{7}x - \frac{2}{3}x^2 + 2) \)
(ii) \( 3 - (4 + x^2 - y^2) + (9x^2 - 9y^2 + 7) \)
(iii) \( [7 - 2x + 5y - (x - y)] - [5x + 3y - 7] \)
Answer:
(i) Open the brackets and invert the signs of the second expression:
\( = \frac{1}{3}x^2 - \frac{4}{7}x + 5 - \frac{2}{7}x + \frac{2}{3}x^2 - 2 \)
Combine the like terms:
\( = \left(\frac{1}{3} + \frac{2}{3}\right)x^2 + \left(-\frac{4}{7} - \frac{2}{7}\right)x + (5 - 2) \)
\( = \frac{3}{3}x^2 - \frac{6}{7}x + 3 \)
\( = x^2 - \frac{6}{7}x + 3 \)

(ii) Remove the parentheses carefully:
\( = 3 - 4 - x^2 + y^2 + 9x^2 - 9y^2 + 7 \)
Group like terms:
\( = (-x^2 + 9x^2) + (y^2 - 9y^2) + (3 - 4 + 7) \)
\( = 8x^2 - 8y^2 + 6 \)

(iii) First, simplify the inner round bracket:
\( 7 - 2x + 5y - (x - y) = 7 - 2x + 5y - x + y = -3x + 6y + 7 \)
Now, subtract the second bracket:
\( (-3x + 6y + 7) - (5x + 3y - 7) \)
\( = -3x + 6y + 7 - 5x - 3y + 7 \)
\( = (-3 - 5)x + (6 - 3)y + (7 + 7) \)
\( = -8x + 3y + 14 \)
In simple words: Work from the innermost brackets outward, changing signs whenever a minus comes before a bracket.

Exam Tip: In nested brackets like \([ \dots ( \dots ) \dots ]\), always resolve the innermost brackets first.

Question 10. What must be added to \( (3x^2 - 7xy + 5y^2) \) to make the sum equal to \( x^2 + 4xy - 2y^2 \)
Answer:
Let the required expression be \( P \).
According to the question:
\( (3x^2 - 7xy + 5y^2) + P = x^2 + 4xy - 2y^2 \)

To find \( P \), subtract the given expression from the required sum:

\( \implies P = (x^2 + 4xy - 2y^2) - (3x^2 - 7xy + 5y^2) \)
\( = x^2 + 4xy - 2y^2 - 3x^2 + 7xy - 5y^2 \)
Group the like terms:
\( = (1 - 3)x^2 + (4 + 7)xy + (-2 - 5)y^2 \)
\( = -2x^2 + 11xy - 7y^2 \)
(or written as \( -2x^2 - 7y^2 + 11xy \))
In simple words: Subtract the starting expression from the final sum to find the missing part.

Exam Tip: If you are unsure what operation to perform, use simple numbers: to find what to add to 3 to get 5, do \( 5 - 3 \).