GSEB Class 7 Maths Solutions Chapter 12 Algebraic Expressions Exercise 12.1

Question 1. Get the algebraic expressions in the following cases using variables, constants and arithmetic operations.
(i) Subtraction of z from y.
(ii) One-half of the sum of numbers x and y.
(iii) The number z multiplied by itself.
(iv) One-fourth of the product of numbers p and q.
(v) Numbers x and y both squared and added.
(vi) Number 5 added to three times the product of numbers m and n.
(vii) Product of numbers y and z subtracted from 10.
(viii) Sum of numbers a and b subtracted from their product.
Answer:
(i) The expression for subtracting z from y is \( y - z \).
(ii) One-half of the sum of x and y can be written as \( \frac { 1 }{ 2 }(x + y) \).
(iii) The number z multiplied by itself is represented as \( z^2 \).
(iv) One-fourth of the product of numbers p and q is \( \frac { 1 }{ 4 }pq \).
(v) Numbers x and y both squared and then added is \( x^2 + y^2 \).
(vi) Number 5 added to three times the product of m and n is \( 5 + 3mn \).
(vii) The product of y and z subtracted from 10 is \( 10 - yz \).
(viii) The sum of a and b subtracted from their product is \( ab - (a + b) \).
In simple words: We are changing written phrases into math symbols and letters. For example, "subtract z from y" simply means writing \( y - z \).

Exam Tip: Pay close attention to keywords like "from", "product", "sum", "one-half", and "squared" as they dictate the order and type of arithmetic operations.

Question 2. (i) Identify the terms and their factors in the following expressions. Show the terms and factors by tree diagrams.
(a) x - 3
(b) 1 + x + x²
(c) y - y³
(d) 5xy² + 7x²y
(e) -ab + 2b² – 3a²
Answer:
(a) \( x - 3 \)

Exam Tip: Remember that a negative sign in front of a term makes -1 one of its factors. For variables raised to a power, like \( x^2 \), list each individual variable as a factor (e.g., \( x \), \( x \)).

Question 2. (ii) Identify terms and factors in the expressions given below:
(a) -4x + 5
(b) -4x + 5y
(c) 5y + 3y²
(d) xy + 2x²y²
(e) pq + q
(f) 1.2ab - 2.4b + 3.6a
(g) \( \frac { 3 }{ 4 }x + \frac { 1 }{ 4 } \)
(h) \( 0.1p^2 + 0.2q^2 \)
Answer:

Exam Tip: Always remember that a constant (a number without any variables) is a term, and it is also its own factor. Be careful with negative signs—they are part of the number factor.

Question 3. Identify the numerical coefficients of terms (other than constants) in the following expressions:
(i) 5 – 3t²
(ii) 1 + t + t² + t³
(iii) x + 2xy + 3y
(iv) 100m + 1000n
(v) -p²q² + 7pq
(vi) 1.2a + 0.8b
(vii) 3.Mr²
(viii) 2(l + b)
(ix) 0.1 y + 0.01 y²
Answer:

Exam Tip: Remember to include the sign (positive or negative) with the numerical coefficient. For terms like \( x \) or \( t^2 \), where no number is explicitly shown, the coefficient is always 1.

Question 4. (a) Identify terms which contain x and give the coefficient of x.
(i) yx² + y
(ii) 13y² – 8yx
(iii) x + y + 2
(iv) 5 + z + zx
(v) 1 + x + xy<
(vi) 12xy² + 25
(vii) 7x + xy²
Answer:

Exam Tip: When asked for the coefficient of a specific variable (like \( x \)), treat any other variables in that term as part of the coefficient, along with the numerical part. For example, in \( 5xy \), the coefficient of \( x \) is \( 5y \).

Question 4. (b) Identify terms which contain y² and give the coefficient of y².
(i) 8 – xy²
(ii) 5y² + 7x
(iii) 2x²y – 15xy² + 7y²
Answer:

Exam Tip: Be careful to identify all terms that contain the specified variable (here \( y^2 \)) and extract their respective coefficients, including any signs. If \( y^2 \) is part of a larger term (e.g., \( -15xy^2 \)), the coefficient is all parts of that term except \( y^2 \).

Question 5. Classify into monomials, binomials and trinomials.
(i) 4y – 7z
(ii) y²
(iii) x + y - xy
(iv) 100
(v) ab - a - b
(vi) 5 – 3t
(vii) 4p²q – 4pq²
(viii) 7mn
(ix) z² - 3z + 8
(x) a² + b²
(xi) z² + z
(xii) 1 + x + x²
Answer:
(i) The expression \( 4y – 7z \) has 2 terms (4y and -7z). Therefore, it is a binomial.
(ii) The expression \( y^2 \) has only 1 term (\( y^2 \)). Therefore, it is a monomial.
(iii) The expression \( x + y - xy \) has 3 terms (x, y and -xy). Therefore, it is a trinomial.
(iv) The expression \( 100 \) has only 1 term (100). Therefore, it is a monomial.
(v) The expression \( ab - a - b \) has 3 terms (ab, -a and -b). Therefore, it is a trinomial.
(vi) The expression \( 5 – 3t \) has two terms (5 and -3t). Therefore, it is a binomial.
(vii) The expression \( 4p^2q – 4pq^2 \) has 2 terms (\( 4p^2q \) and \( -4pq^2 \)). Therefore, it is a binomial.
(viii) The expression \( 7mn \) has only one term (7mn). Therefore, it is a monomial.
(ix) The expression \( z^2 - 3z + 8 \) has 3 terms (\( z^2 \), -3z and +8). Therefore, it is a trinomial.
(x) The expression \( a^2 + b^2 \) has 2 terms (\( a^2 \) and \( b^2 \)). Therefore, it is a binomial.
(xi) The expression \( z^2 + z \) has 2 terms (\( z^2 \) and \( z \)). Therefore, it is a binomial.
(xii) The expression \( 1 + x + x^2 \) has 3 terms (1, x and \( x^2 \)). Therefore, it is a trinomial.
In simple words: We classify algebraic expressions based on how many "terms" they have. An expression with one term is called a "monomial". If it has two terms, it's a "binomial". If it has three terms, it's a "trinomial".

Exam Tip: To count terms correctly, remember that terms are separated by plus (+) or minus (-) signs. Make sure to simplify the expression first if possible, before counting terms.

Question 6. State whether a given pair of terms is of like or unlike terms.
(i) 1, 100
(ii) -7x, \( \frac {5}{ 2 }x \)
(iii) -29x, -29y
(iv) 14xy, 42yx
(v) 4m²p, 4mp²
(vi) 12xz, 12x²z²
Answer:
(i) 1, 100 is a pair of like terms because both are constants (can be written as \( 1x^0 \) and \( 100x^0 \)).
(ii) \( -7x \), \( \frac {5}{ 2 } x \) is a pair of like terms because they both have the same variable part, which is \( x \).
(iii) \( -29x \), \( -29y \) is a pair of unlike terms because their variable parts (\( x \) and \( y \)) are different.
(iv) \( 14xy \), \( 42yx \) is a pair of like terms because the order of variables does not change the term (xy is the same as yx).
(v) \( 4m^2p \), \( 4mp^2 \) is a pair of unlike terms because the powers of the variables are different (\( m^2p \) is not the same as \( mp^2 \)).
(vi) \( 12xz \), \( 12x^2z^2 \) is a pair of unlike terms because the powers of the variables are different (\( xz \) is not the same as \( x^2z^2 \)).
In simple words: Like terms have the exact same letters (variables) and those letters must have the same small power numbers. If the letters or their powers are different, they are "unlike terms".

Exam Tip: For terms to be "like terms", the variables and their exponents must be identical. The numerical coefficients can be different, and the order of variables in a term doesn't matter (e.g., \( xy \) is the same as \( yx \)).

Question 7. Identify like terms in the following:
(a) – xy², – 4yx², 8x², 2xy², 7y, – 11x², – 100x, – 11yx, 20x²y, – 6x², y, 2xy, 3x
(b) 10qp, 7p, 8q, – p²q², – 7qp, – 100q, – 23, 12q²p², – 5p², 41, 2405p, 78qp, 13q²p, qp², 701p²
Answer:
(a) The like terms are:
\( -xy^2 \) and \( 2xy^2 \)
\( -4yx^2 \) and \( 20x^2y \)
\( 8x^2 \), \( -11x^2 \) and \( -6x^2 \)
\( 7y \) and \( y \)
\( -100x \) and \( 3x \)
\( -11yx \) and \( 2xy \)
(b) The like terms are:
\( 10qp \), \( -7qp \) and \( 78qp \)
\( 7p \) and \( 2405p \)
\( 8q \) and \( -100q \)
\( -p^2q^2 \) and \( 12q^2p^2 \)
\( -23 \) and \( 41 \)
\( -5p^2 \) and \( 701p^2 \)
\( 13q^2p \) and \( qp^2 \)
In simple words: To find like terms, you just need to match the variable parts exactly. The numbers in front (coefficients) don't matter. For example, \( xy^2 \) and \( 2xy^2 \) are like terms because both have \( xy^2 \).

Exam Tip: When identifying like terms, remember that the order of multiplication for variables does not affect whether terms are alike (e.g., \( yx^2 \) is the same as \( x^2y \)). Pay close attention to the exponents of each variable.