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State Board · Class 7 · Mathematics

Lesson 8: Algebraic Expressions

5 concepts · 40 practice questions with answers · Free to read

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Terms, coefficients, variables and like terms

From daily life

A pen costs 7 rupees. 3 pens cost 7 × 3 and 10 pens cost 7 × 10. When the number of pens keeps changing, we write a letter n in its place and the cost becomes one expression, 7n rupees.

Simple definition

A letter used for a number that changes is called a variable. An arrangement of numbers, variables and the operations + − × ÷ is called an algebraic expression, such as 7n or 2n + 1. The pieces into which the + and − signs cut an expression are called terms, and the number in front of the variable in a term is its coefficient.

12 × 1 + 1 = 322 × 2 + 1 = 532 × 3 + 1 = 742 × 4 + 1 = 92 × n + 1

In the figure: 1 triangle needs 3 sticks, 2 triangles need 5, 3 triangles need 7 and 4 triangles need 9 sticks. Each new triangle adds 2 sticks, so n triangles need 2n + 1 sticks.

Rule 1: Write the changing number as a letter to form an expression — Put a letter in place of what changes and write the given operations around that letter as they are.

  • Easy: If one pen costs 7 rupees, then n pens cost 7n rupees.
  • Medium: In the row of triangles, if n = 6, the sticks 2n + 1 are 2 × 6 + 1 = 13.
  • Exam type: A boy had x rupees and spent 15 rupees. The money left is x − 15. If x = 40, then 40 − 15 = 25 rupees are left.
6a2 − 3b + 5123coefficient 6variable acoefficient −3variable bconstant term53 terms in all

In the figure: The expression 6a² − 3b + 5 has 3 terms: term 1 has coefficient 6, term 2 has coefficient −3 and term 3 is the constant term 5. The 2 in a² is the power.

Rule 2: Identify the term, the coefficient and the constant term — The number in front of the variable is the coefficient. If only x is written the coefficient is 1, and for −x it is −1. A term that is only a number, with no variable, is called the constant term.

  • Easy: In 9m the coefficient is 9 and the variable is m.
  • Medium: 5 − k has 2 terms: the constant term 5 and the term −k. The coefficient of k is −1.
  • Exam type: 8p² − p + 12 has 3 terms. The coefficients are 8 and −1, and 12 is the constant term.

Rule 3: Like terms carry identical letters raised to identical powers — The coefficients may differ. Even if the order of the variables changes (xy and yx), the terms are still like terms.

  • Easy: 4x and 9x are like terms, so 4x + 9x = (4 + 9)x = 13x.
  • Medium: 6xy and yx are like terms. But 4x and 4x² are not like terms, because the powers differ.
  • Exam type: Among 7m, 2n, −3m and m², only 7m and −3m are like terms. 2n has a different variable and m² has a different power.
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Terms, coefficients, variables and like terms

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Remember

The + or − sign in an expression stays with the term that follows it. So in 5 − k the second term is −k.

Common mistake

Treating x and x² as like terms. Even if the variable is the same, terms with different powers are not like terms.

Shortcut

Look at two terms and ask, "Same variables? Same powers?" Only if both answers are yes are the terms like terms.

Spot the difference

TermCoefficientVariable
9m9m
−x−1x
2ab2a, b
55no variable (constant term)

Practice questions

Easy
  1. 1How many terms does 7p² − 3p + 9 have? What is the coefficient of p²?
  2. 2What is the coefficient of x in −x + 4?
  3. 3Choose the like terms of 5ab: 3ba, 5a, −2ab, ab².
Medium
  1. 4In a row of triangles, n triangles need 2n + 1 sticks. If n = 8, how many sticks are needed?
  2. 5Give the coefficient of each term of 4a − 9b + 5ab − 7 and identify the constant term.
  3. 6Group the like terms among 3x, 2y, −5x, 4y, x².
Challenging
  1. 7A notebook costs x rupees and a pen costs 4 rupees. Write the cost of 3 notebooks and 5 pens as an expression. If x = 12, what is the cost?
  2. 8Which are the groups of like terms in 9m² − 2m + m² + 5m? Write the sum of each group.
Show answers (try them yourself first)
  1. 13 terms; the coefficient of p² is 7. 2) The coefficient is −1. 3) 3ba and −2ab.
  2. 42 × 8 + 1 = 17 sticks. 5) The coefficients are 4, −9 and 5; the constant term is −7. 6) 3x and −5x; 2y and 4y; x² is alone.
  3. 7The cost is 3x + 5 × 4 = 3x + 20 rupees. If x = 12, then 3 × 12 + 20 = 36 + 20 = 56 rupees.
  4. 89m² and m² form one group: 9m² + m² = 10m². −2m and 5m form the other group: −2m + 5m = 3m.

Addition and subtraction of algebraic expressions

From daily life

One bag has 2 apples and 3 guavas, another bag has 4 apples and 1 guava. When the bags are combined, apples are counted with apples and guavas with guavas. Expressions are added in the same way.

Simple definition

To add like terms, add their coefficients and keep the variable as it is. Unlike terms do not combine. To subtract, reverse the sign on each term of the second expression and then add.

xxyyy2x + 3yxxxxy+4x + yxxxxxxyyyy=6x + 4y6x + 4y

In the figure: One bag has 2x + 3y and the other has 4x + y. There are 2 + 4 = 6 pieces of x and 3 + 1 = 4 pieces of y. So (2x + 3y) + (4x + y) = 6x + 4y.

Rule 1: Adding like terms — add the coefficients — Keep the variable as it is and only add or subtract the coefficients.

  • Easy: 6a + 5a = (6 + 5)a = 11a.
  • Medium: 9m − 4m = (9 − 4)m = 5m and −7k + 3k = (−7 + 3)k = −4k.
  • Exam type: 5x − 8x + 2x = (5 − 8 + 2)x = −x.

Rule 2: Adding two expressions — collect the like terms — Remove the brackets, write the like terms together and add each group.

  • Easy: (3a + 5b) + (2a + 4b) = (3a + 2a) + (5b + 4b) = 5a + 9b.
  • Medium: (3x + 4) + (5x − 2) = (3x + 5x) + (4 − 2) = 8x + 2.
  • Exam type: (7x² + 3x) + (2x² − 5x) = (7x² + 2x²) + (3x − 5x) = 9x² − 2x.
(7a + 5b) − (3a − 2b)change every sign in the second bracket7a + 5b − 3a + 2b4a + 7b7a − 3a = 4a · 5b + 2b = 7b

In the figure: (7a + 5b) − (3a − 2b) = 7a + 5b − 3a + 2b = 4a + 7b. When every sign in the second bracket is changed, −2b becomes +2b.

Rule 3: Subtraction — change every sign in the second bracket — If a − sign stands before a bracket, reverse the signs of all terms inside it and then add.

  • Easy: (9a + 4) − (3a + 1) = 9a + 4 − 3a − 1 = 6a + 3.
  • Medium: (3x + 4) − (5x − 2) = 3x + 4 − 5x + 2 = −2x + 6.
  • Exam type: (6m − 2n) − (m − 7n) = 6m − 2n − m + 7n = 5m + 5n.
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Addition and subtraction of algebraic expressions

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Rule 4: Unlike terms cannot be added; check the answer — Write unlike terms as they are. To see whether the answer is right, put in one value of x and compare both sides.

  • Easy: In 3x + 7y the terms are unlike, so the sum stays 3x + 7y.
  • Medium: (5a + 2b) + (3a + 4c) = 8a + 2b + 4c, because 2b and 4c do not combine.
  • Exam type: (2x + 5) + (4x − 1) = 6x + 4. Take x = 3: (2 × 3 + 5) + (4 × 3 − 1) = 11 + 11 = 22 and 6 × 3 + 4 = 22; both are equal.
Remember

In addition only the coefficients are added and the variable stays the same. In subtraction every sign in the second bracket is reversed.

Common mistake

Adding unlike terms: 3x + 7y = 10xy. x and y are different variables, so the answer stays 3x + 7y.

Shortcut

In the vertical method, write like terms one below the other. Then add or subtract the coefficients in each column.

Spot the difference

ExpressionLike terms?Sum
5x + 2xYes7x
5x + 2yNo5x + 2y
5x + 2x²No5x + 2x²
5xy + 2yxYes7xy

Practice questions

Easy
  1. 1Find the sum 8a + 5a.
  2. 2Add (4x + 3) + (2x + 6).
  3. 3Subtract (9y + 2) − (4y + 2).
Medium
  1. 4Add (5a + 3b) + (2a − 7b).
  2. 5Subtract (2m − 5n) from (8m − 3n).
  3. 6Add 3x² − 4x + 5 and x² + 6x − 2.
Challenging
  1. 7Subtract (3p + q − 6) from (7p − 2q + 4).
  2. 8One bag has (3a + 2b) rupees and another has (a + 5b) rupees. What is the total amount? What do you get if the second bag's amount is subtracted from the first?
Show answers (try them yourself first)
  1. 18a + 5a = 13a. 2) (4x + 3) + (2x + 6) = 6x + 9. 3) (9y + 2) − (4y + 2) = 9y + 2 − 4y − 2 = 5y.
  2. 4(5a + 3b) + (2a − 7b) = 7a − 4b. 5) (8m − 3n) − (2m − 5n) = 8m − 3n − 2m + 5n = 6m + 2n.
  3. 6(3x² − 4x + 5) + (x² + 6x − 2) = 4x² + 2x + 3.
  4. 7(7p − 2q + 4) − (3p + q − 6) = 7p − 2q + 4 − 3p − q + 6 = 4p − 3q + 10.
  5. 8The total is (3a + 2b) + (a + 5b) = 4a + 7b. The difference is (3a + 2b) − (a + 5b) = 3a + 2b − a − 5b = 2a − 3b.

Multiplication of algebraic expressions

From daily life

To get the area of a rectangular plot we multiply its length and breadth. If the length and breadth are in letters, the product is also in letters, and it is like finding the area of each part separately and adding them.

Simple definition

For a product of two monomials, begin with the coefficients (keeping their signs) and then multiply the variables. To multiply a monomial by a binomial, multiply every term of the binomial by the monomial. To multiply two binomials, multiply every term by every term and combine the like terms.

Rule 1: Monomial × monomial — coefficients first, then variables — Multiply the coefficients with their signs. Multiplying the same variable raises its power: x × x = x².

  • Easy: 4a × 5b = 4 × 5 × a × b = 20ab.
  • Medium: (−3x) × 6y² = −3 × 6 × x × y² = −18xy².
  • Exam type: (−2m²) × (−5mn) = (−2) × (−5) × m² × m × n = 10m³n.
3x12x433 × x + 3 × 4 = 3x + 12

In the figure: The rectangle with sides 3 and x + 4 has two parts, 3x and 12. So 3(x + 4) = 3x + 12.

Rule 2: Monomial × binomial — multiply every term — Multiply the monomial with each term and take care of the signs.

  • Easy: 2(x + 5) = 2 × x + 2 × 5 = 2x + 10.
  • Medium: x(x + 6) = x × x + x × 6 = x² + 6x.
  • Exam type: (5a − 2b) × 3c = 5a × 3c − 2b × 3c = 15ac − 6bc.
x23x2x6x3x2x2 + 3x + 2x + 6 = x2 + 5x + 6

In the figure: The rectangle with sides x + 3 and x + 2 has four parts, x², 3x, 2x and 6. Their sum is (x + 3)(x + 2) = x² + 3x + 2x + 6 = x² + 5x + 6.

Rule 3: Binomial × binomial — make four products, then combine like terms — Multiply each term of the first bracket by each term of the second bracket and then combine the like terms.

  • Easy: (x + 1)(x + 4) = x² + 4x + x + 4 = x² + 5x + 4.
  • Medium: (2x + 3)(x − 1) = 2x² − 2x + 3x − 3 = 2x² + x − 3.
  • Exam type: (a + 2b)(3a − b) = 3a² − ab + 6ab − 2b² = 3a² + 5ab − 2b².
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Multiplication of algebraic expressions

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Rule 4: Area and checking — If the sides are expressions, area = length × breadth. To see whether the answer is right, put in one value of x and compare both sides.

  • Easy: Take x = 4: (4 + 3) × (4 + 2) = 7 × 6 = 42 and 4² + 5 × 4 + 6 = 16 + 20 + 6 = 42. So (x + 3)(x + 2) = x² + 5x + 6 is correct.
  • Medium: If a rectangle is (3x + 2) metres long and (x + 1) metres wide, its area is (3x + 2)(x + 1) = 3x² + 3x + 2x + 2 = 3x² + 5x + 2 square metres.
  • Exam type: In the rectangle above, if x = 4, the length is 14 and the breadth is 5 metres; the area is 14 × 5 = 70. The formula also gives 3 × 4² + 5 × 4 + 2 = 48 + 20 + 2 = 70.
Remember

In multiplication, first multiply the coefficients (with signs) and then the variables. x × x = x², but x + x = 2x.

Common mistake

Multiplying only the first and the last terms in binomial × binomial: (x + 3)(x + 2) = x² + 6. Do not forget the two middle products.

Shortcut

Draw the four parts of the rectangle and add them. Or put x = 1 and see: both sides must give the same answer.

Spot the difference

Type of multiplicationExampleAnswer
Monomial × monomial4a × 5b20ab
Same variablex × xx²
Monomial × binomial2(x + 5)2x + 10
Binomial × binomial(x + 1)(x + 4)x² + 5x + 4

Practice questions

Easy
  1. 1Multiply 3a × 7b.
  2. 2Multiply (−4x) × 5y.
  3. 3Multiply 3(x + 7).
Medium
  1. 4Multiply (−2x²) × 6x.
  2. 5Multiply y(y − 5).
  3. 6Multiply (x + 2)(x + 6).
Challenging
  1. 7Multiply (2a − 3)(a + 4).
  2. 8A rectangle is (2x + 1) metres long and (x + 3) metres wide. What is its area? Check the answer with x = 2.
Show answers (try them yourself first)
  1. 13a × 7b = 21ab. 2) (−4x) × 5y = −20xy. 3) 3(x + 7) = 3x + 21.
  2. 4(−2x²) × 6x = −12x³. 5) y(y − 5) = y² − 5y. 6) (x + 2)(x + 6) = x² + 6x + 2x + 12 = x² + 8x + 12.
  3. 7(2a − 3)(a + 4) = 2a² + 8a − 3a − 12 = 2a² + 5a − 12.
  4. 8The area is (2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3 square metres. Take x = 2: (2 × 2 + 1) × (2 + 3) = 5 × 5 = 25 and 2 × 2² + 7 × 2 + 3 = 8 + 14 + 3 = 25.

Types of expressions

From daily life

The number of items on a shop bill shows whether the bill is small or big. In the same way, the number of terms in an expression decides its name.

Simple definition

An expression with 1 term is a monomial, with 2 terms a binomial, with 3 terms a trinomial, and with more than three terms a polynomial. The name depends on the number of terms, not on the power.

1Monomial9m2Binomial4a−7b3Trinomialx23x−104Polynomialy3−2y25y−6

In the figure: 9m with 1 term is a monomial, 4a − 7b with 2 terms is a binomial, x² + 3x − 10 with 3 terms is a trinomial and y³ − 2y² + 5y − 6 with 4 terms is a polynomial.

Rule 1: Cut the expression at the + and − signs and count the terms — A − sign at the very start of an expression is part of the first term. Name the expression by its number of terms.

  • Easy: 8k has 1 term, so it is a monomial.
  • Medium: 5p − 2q has 2 terms, so it is a binomial.
  • Exam type: x² + 5x + 6 has 3 terms, so it is a trinomial.
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Types of expressions

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Rule 2: First combine like terms, then count the terms — After simplifying, the number of terms can become smaller and the name changes too.

  • Easy: 3k + 2k = 5k. There were 2 terms, now there is 1; it is a monomial.
  • Medium: 6a + 2b − 4a = 2a + 2b. There were 3 terms; after simplifying there are 2; it is a binomial.
  • Exam type: x² + 2x + 3x − 7 = x² + 5x − 7. There were 4 terms; after simplifying there are 3; it is a trinomial.

Rule 3: A power or a plain number does not change the name — A plain number is also one term, and even a large power can sit in a single term.

  • Easy: 9 is one term made only of a number, so it is a monomial.
  • Medium: x⁵ has power 5, yet it is a single term, so it is a monomial.
  • Exam type: y³ − 2y² + 5y − 8 has 4 terms, so it is a polynomial.
Remember

The name of an expression depends on the number of its terms, not on how large the power is.

Common mistake

Calling x² a binomial because the power is 2. It has only 1 term, so it is a monomial.

Shortcut

Count the + and − signs between terms and add 1; do not count a − at the very start. That gives the number of terms.

Spot the difference

NameNumber of termsExample
Monomial19m
Binomial24a − 7b
Trinomial3x² + 3x − 10
Polynomial4 or morey³ − 2y² + 5y − 6

Practice questions

Easy
  1. 1What is the name of the expression 6y?
  2. 2What is the name of the expression 3a − 8?
  3. 3What is the name of the expression x² + 2x + 9?
Medium
  1. 4Write 2m + 5m − 3 in simplest form and name it.
  2. 5Write 4a + 6b − 2a − 6b in simplest form and name it.
  3. 6What kind of expression is y⁴?
Challenging
  1. 7How many terms does p³ − 5p² + 2p − 1 have, and what is its name?
  2. 8Write 3x² + 4x + 2x² − 4x + 7 in simplest form and name it.
Show answers (try them yourself first)
  1. 11 term; monomial. 2) 2 terms; binomial. 3) 3 terms; trinomial.
  2. 42m + 5m − 3 = 7m − 3; 2 terms, so a binomial. 5) 4a + 6b − 2a − 6b = 2a; 1 term, so a monomial.
  3. 6y⁴ has power 4 but only 1 term, so it is a monomial.
  4. 74 terms; polynomial.
  5. 83x² + 4x + 2x² − 4x + 7 = 5x² + 7; 2 terms, so a binomial.

Equations in one variable

From daily life

If the two pans of a balance are level and you take some weight out of one pan, you must take the same weight out of the other pan too. Only then does the balance stay level. The two sides of an equation behave the same way.

Simple definition

Two expressions joined by an = sign make an equation. The value of the variable for which both sides become equal is the solution of the equation. If you do the same operation (add, subtract, multiply or divide) on both sides, the equation stays true.

xx=2x + 3 = 11xx=2x = 8− 3 (on both sides)x=x = 4÷ 2 (on both sides)

In the figure: 2x + 3 = 11 ⇒ 2x = 8 ⇒ x = 4. When 3 is taken from both pans, 2x and 8 are left. Then dividing both sides by 2 gives the value 4 for x.

Rule 1: Add or subtract the same number on both sides — To remove the number next to the variable, do the opposite operation on both sides.

  • Easy: x + 6 = 15 ⇒ x + 6 − 6 = 15 − 6 ⇒ x = 9.
  • Medium: y − 4 = 11 ⇒ y − 4 + 4 = 11 + 4 ⇒ y = 15.
  • Exam type: 3x − 5 = 16 ⇒ 3x − 5 + 5 = 16 + 5 ⇒ 3x = 21 ⇒ x = 7.
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Equations in one variable

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Rule 2: Multiply or divide both sides by the same number — If the variable has a coefficient, divide both sides by that coefficient.

  • Easy: 4x = 28 ⇒ 4x ÷ 4 = 28 ÷ 4 ⇒ x = 7.
  • Medium: x ÷ 5 = 6 ⇒ x ÷ 5 × 5 = 6 × 5 ⇒ x = 30.
  • Exam type: 5y + 2 = 27 ⇒ 5y = 25 ⇒ y = 5. First 2 was subtracted and then both sides were divided by 5.

Rule 3: If both sides have the variable, a term changes sides and changes its sign — Take the terms with the variable to one side and the numbers to the other. When a term crosses over, its sign is reversed.

  • Easy: 5x = 2x + 12 ⇒ 5x − 2x = 12 ⇒ 3x = 12 ⇒ x = 4.
  • Medium: 7y − 3 = 3y + 9 ⇒ 7y − 3y = 9 + 3 ⇒ 4y = 12 ⇒ y = 3.
  • Exam type: 3(x + 2) = 2x + 11 ⇒ 3x + 6 = 2x + 11 ⇒ 3x − 2x = 11 − 6 ⇒ x = 5.

Rule 4: Form an equation from a word problem and check the answer — Give a letter to the unknown number, write an equation from the sentence, solve it and put the answer back into the problem.

  • Easy: Adding 9 to a number gives 25. x + 9 = 25 ⇒ x = 16. Check: 16 + 9 = 25.
  • Medium: A rectangle is 4 cm longer than it is wide and its perimeter is 28 cm. Taking the width as x: 2(x + x + 4) = 28 ⇒ 4x + 8 = 28 ⇒ x = 5. The width is 5 cm and the length is 9 cm; check: 2 × (5 + 9) = 28.
  • Exam type: Sayli has x marbles and Rohan has 7 more. Together they have 51 marbles, so x + (x + 7) = 51 ⇒ 2x + 7 = 51 ⇒ x = 22. Sayli has 22 and Rohan has 29; check: 22 + 29 = 51.
Remember

Always do the same operation on both sides. Then the equation stays true and the solution does not change.

Common mistake

Not changing the sign when a term changes sides: x + 5 = 12 ⇒ x = 12 + 5. The correct step is x + 5 = 12 ⇒ x = 12 − 5 ⇒ x = 7.

Shortcut

When you get the answer, put it into the original equation. If the left side and the right side are equal, the answer is correct.

Spot the difference

OperationEquationSolution
Subtract from both sidesx + 6 = 15x = 9
Add to both sidesy − 4 = 11y = 15
Divide both sides4x = 28x = 7
Move a term to the other side5x = 2x + 12x = 4

Practice questions

Easy
  1. 1Solve x + 8 = 20.
  2. 2Solve y − 7 = 5.
  3. 3Solve 6m = 42.
Medium
  1. 4Solve 3x + 4 = 19.
  2. 5Solve x ÷ 4 = 7.
  3. 6Solve 9a − 2 = 4a + 13.
Challenging
  1. 7Solve 2(y + 3) = 3y − 1 and check the answer.
  2. 8Three times a number plus 7 gives 31. Which number is it?
Show answers (try them yourself first)
  1. 1x + 8 = 20 ⇒ x + 8 − 8 = 20 − 8 ⇒ x = 12. 2) y − 7 = 5 ⇒ y − 7 + 7 = 5 + 7 ⇒ y = 12. 3) 6m = 42 ⇒ 6m ÷ 6 = 42 ÷ 6 ⇒ m = 7.
  2. 43x + 4 = 19 ⇒ 3x = 15 ⇒ x = 5. 5) x ÷ 4 = 7 ⇒ x ÷ 4 × 4 = 7 × 4 ⇒ x = 28.
  3. 69a − 2 = 4a + 13 ⇒ 9a − 4a = 13 + 2 ⇒ 5a = 15 ⇒ a = 3.
  4. 72(y + 3) = 3y − 1 ⇒ 2y + 6 = 3y − 1 ⇒ 6 + 1 = 3y − 2y ⇒ y = 7. Check: 2 × (7 + 3) = 20 and 3 × 7 − 1 = 20.
  5. 83x + 7 = 31 ⇒ 3x = 24 ⇒ x = 8. Check: 3 × 8 + 7 = 31.