Quantitative Aptitude - SPLessons

Simplifying Algebraic Expressions

Chapter 53

SPLessons 5 Steps, 3 Clicks
5 Steps - 3 Clicks

Simplifying Algebraic Expressions

shape Introduction

An equivalent expression is found that is simpler than the original when an algebraic expression is simplified. Algebraic expressions contain alphabetic symbols as well as the numbers. By simplifying, the value of the expression does not change.

In this chapter, the concepts like algebra, expressions, key definitions and simplifying algebraic expressions are explained in detail.


shape Methods

What is Algebra?

Now, to make the question less open-ended, what does it mean to say 3x + 7x = 10x

This is the type of algebraic statement which would have seen in school and most of the students will accept the fact that 3x + 7x = 10x.

But does it mean to say?

Well, first recognize that the “x’s” here are serving as place orders that represent numbers.

In this case, they are representing every number imaginable. So, 3x + 7x has no particular value and neither does 10x.

The value of 3x + 7x and 10x depends on the value of x.

If a value is assigned to x,

Say x = 2

LHS → 3x + 7x = 3 * 2 + 7 * 2 = 6 + 14 = 20

RHS → 10x = 10 * 2 = 20

LHS = RHS if x = 2


Similarly Say x = 17.3

LHS → 3x + 7x = 3 * 17.3 + 7 * 17.3 = 173

RHS → 10x = 10 * 17.3 = 173

LHS = RHS if x =17.3

From here, a general comment about 3x + 7x and 10x is “For every number, x, that exists, 3x + 7x = 10x”.

This kind of generalization is what Algebra is all about.


Algebra: Branch of mathematics in which symbols represent quantities and express general relationships that hold for all numbers. In other words, algebra is essentially a tool that uses better to express the general relationships that hold true for all numbers.

If (x + 2)(x + 7) = \(x^2\) + 9x + 14, it means that for every possible value of x, (x + 2)(x + 7) and (x + 2)(x + 7) will have the same evaluation.

For example, if x = 3

LHS → (3 + 2) (3 + 7) = 5 * 10 = 50

RHS → \(3^2\) + 9 * 3 + 14 = 9 + 27 + 14 = 50

So, LHS = RHS


Key definitions

Variable

Variable is any symbol (typically a letter) that stands for either a single number or all existing numbers.

Examples: n, x, k


Constant

Constant is either a number on its own or a letter/symbol that represents a value that doesn’t change.

Examples: 6, 7.2, π


Term

Term can be a constant on its own or the product (or quotient) of a constant and 1 or more variables.

Examples: 14, 5x, 8\(xy^3\), \(\frac{jk}{5m^3}\)


Coefficient

Coefficient is the constant of a term.

Examples: 6 is the coefficient of 6x

\(\frac{2}{3}\) is the coefficient of \(\frac{2y}{3}\)


Expression

An expression is a collection of one or more terms combined using addition and/or subtraction.

Examples:

\(w^3\) – \(3^2\) + 5\(y\)

x – 1

\(\frac{2x^4}{5}[latex] + [latex]\frac{1}{y^3}[latex] – [latex]5x^2y \)+ \(x\) – \( 3y\) + 9


Monomial

Monomial ia an expression with 1 term.

Examples: 14, 5x, 8\( xy^3\), \(\frac{jk}{5m^3}\)


Binomial

Binomial is an expression with 2 terms.

Examples:

\(x^2\) + 3\(y\)

w – 8


Polynomial

In general, polynomial is an expression with 1 or more terms.


Simplifying expressions

Like terms

Like terms are the terms with identical variables.

Examples:

3x and -5x are like terms since the variable in both the terms is ‘x’.

7\(xy^2\) and 4\(y^2x\) are like terms since the variable in both the terms is \(xy^2\).

\(\frac{5k}{m^8}\) and \(\frac{k}{4m^8}\) are like terms since the variable in both the terms is \(\frac{k}{m^8}\).

\(3x^2\) and \(4x^3\) are not like terms since the variables are not same in both the terms.



This process of combining terms is known as simplifying.

To add expressions in brackets, remove the brackets

To subtract the expressions in brackets, add the “opposites”.

Examples:

Simplify the following

1. 3x + 5y – x – 9y

Here, 3x and x, 5y and 9y have the same coefficients

So, 3x + 5y – x – 9y = 2x -4y


2. 3ab – 7b + 2b – 4ba

Here, 3ab and 4ba, 7b and 2b have the same coefficients

So, 3ab – 7b + 2b – 4ba = -ab – 5b


3. (2w – 10x + y) + (9x – y – 7w)

First remove the brackets

Now, 2w – 10x + y + 9x – y – 7w

-5w – x


4. (2w – 10x + y) – (9x – y – 7w)

First remove the brackets

Now, 2w – 10x + y – 9x + y + 7w

9w – 19x + 2y

shape Samples

1. Quantitative Comparison

Quantity A Quantity B
3k + 5 11 – 3k

A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.

Solution:

    Add 3k to both the quantities i.e.

    Quantity A → 3k + 5 + 3k = 6k + 5

    Quantity B → 11 – 3k + 3k = 11

    Subtract 5 from both quantities i.e.

    Quantity A → 6k + 5 – 5 = 6k

    Quantity B → 11 – 5 = 6

    Finally divide both quantities by 6 i.e.

    Quantity A → \(\frac{6k}{6}\) = k

    Quantity B → \(\frac{6}{6}\) = 1

    Given that k > 0

    So, k could be greater than 1 or 1 could be greater than k.

    So, the correct answer is D.


2. \(37^2\) – (23)(37) – 37 – (37)(15) =?
A. -74
B. -37
C. 0
D. 37
E. 74

Solution:

    \(37^2\) – (23)(37) – 37 – (37)(15)

    = (37)(37) – (23)(37) – (1)(37) – (37)(15)

    = -(2) (37)

    = -74

    \(37^2\) – (23)(37) – 37 – (37)(15) = -74

    Option A is the right answer.


3. If \(3^x + 3^x +3^x\) = \(5^x + 5^x + 5^x + 5^x + 5^x\), what is the value of x?
A. 1
B. 3
C. 2
D. -1
E. -3

Solution:

    \(3^x + 3^x +3^x\) = \(5^x + 5^x + 5^x + 5^x + 5^x\)

    3 (\(3^x\)) = 5 (\(5^x\))

    \(3^1 (3^x) \) = \(5^1 (5^x) \)

    \(3^{x + 1}\) = \(5^{x + 1}\)

    Divide with \(5^{x + 1}\) on both sides

    \(\frac{3^{x + 1}}{ 5^{x + 1}}\) = 1

    \((\frac{3}{5})^{x + 1}\) = 1

    x + 1 = 0

    x = -1

    Option D is the right answer.


4. Simplify 7a – 10a by combining like terms.
A. 3a
B. -3a
C. 4a
D. a
E. 0

Solution:

    7a – 10a

    Both terms have the same variable part, a.

    = (7 – 10)a

    = -3a

    Option B is the right answer.


5. Simplify 15b + 9 + 5b – 2by combining like terms.
A. 10b + 2
B. 5b
C. 3
D. 6 + 11b
E. 20b + 7

Solution:

    15b + 9 + 5b – 2

    Two terms have the same variable part, b. The other pair of terms are constant terms that can be combined together.

    = 15b + 5b + 9 – 2

    = (15 + 5)b + 7

    = 20b + 7

    Option E is the right answer.