Algebraic Expression - How to simplify problems

Combine Like Terms

Like terms are terms that have the same variable part i.e. they only differ in their coefficients. Combining like terms is very often required in the process of simplifying equations.

For example:

2x and –5x are like terms

a and are like terms

6x and 5y are unlike terms

Like terms can be added or subtracted from one another.

For example:

a + a = 2 × a = 2a (We usually write 2 × a as 2a)

2a + 4a = 6a

a + a + a = 3a

2a + 4 (Unlike terms cannot be simplified)

4a + 3b (Unlike terms cannot be simplified)

6a – 3a = 3a

8b – 8b = 0

5a – 3 (Unlike terms cannot be simplified)

6a – 4b (Unlike terms cannot be simplified)

Example 1:

Simplify: 8xy – 5yx = 1

Solution:

Step 1: 5yx is the same as 5xy using the

commutative property

Step 2: Since the right side is already simple, we can work on the left side expression:

8xy – 5yx = 8xy – 5xy = 3xy

Putting back the left side and right side of the equation:

3xy = 1

Answer: 3xy = 1

Example 2:

Simplify: 7a + 5b – 6b + 8a + 2b = 0

Solution:

Step 1: Group together the like terms:

7a + 5b – 6b + 8a + 2b = 0
(7a + 8a) + (5b – 6b + 2b) = 0

Step 2: Then simplify:

15a + b = 0

Answer: 15a + b = 0

Multiplication and Division of Terms

The coefficients and variables of terms can be multiplied or divided together in the process of simplifying equations.

For example:

3 × 4b = 3 × 4 × b = 12b

5a × 3a = 5 × a × 3 × a = 5 × 3 × a × a = 15a² (using exponents)

Beware!	a × a = a2
	a + a = 2a

Example 1:

Simplify: 4a × 5a ÷ 2a = 60

Solution:

Step 1: Perform the multiplication and division

Step 2: Isolate variable a

Answer: a = 6

Removal of Brackets - Distributive Property

Sometimes removing brackets (parenthesis) allows us to simplify the expression. Brackets can be removed by using the distributive property. This is often useful in simplifying equations.

For example:

3(a – 3) + 4 = 3 × a + 3 × (-3) + 4 = 3a – 9 + 4 = 3a – 5

5 – 6 (b + 1) = 5 + ( – 6 ) × b + (– 6) × 1 = 5 – 6b – 6 = – 6b – 1

Example 1:

Simplify: 5(a – 4) + 3 = 8

Solution:

Step 1: Remove the brackets

5a – 20 + 3 = 8

Step 2: Isolate variable a

5a = 8 – 3 + 20
5a = 25

Answer: a = 5

Cross Multiplication

Cross multiplication allows you to remove denominators from fractions in an equation. Note that this technique applies only towards simplifying equations, not to simplifying expressions.

For example, if you have the equation:

then you can multiply the numerator of one fraction with the denominator of the other fraction (across the = sign) as shown:

to obtain the equation

(2 × 6) = a × 3

Example 1:

Simplify:

Solution:

Step 1: Cross Multiply

4 × a = 8 × 5
4a = 40

Step 2: Isolate variable a

Answer: a = 10

In the equation 3x – 2 = 16, notice that the variable is being multiplied and has a term being subtracted. To isolate the variable, we need to add 2 to both sides and then divide both sides by 3.

Example:

Solve 3x – 2 = 16

Solution:

3x – 2 = 16

3x – 2 + 2 = 16 + 2 (add 2 to both sides)

3x = 18 (simplify)

x = 6 (divide both sides by 3)

Check:

3x – 2 = 16

3 • 6 – 2 = 16 (Substitute x = 6 into the original equation)

Consider the equation, . To isolate the variable, we need to subtract 2 from both sides and then multiply both sides by 4.

Example:

Solve

Solution:

Check:

Solving multi-step equations with whole numbers

To solve a multi-step equation, we would start by trying to simplify the equation by combining like terms and using the distributive property whenever possible.

Consider the equation 2(x + 1) – x = 5. First, we will use the distributive property to remove the parenthesis and then we can combine like terms and the isolate the variable.

Example:

Solve 2(x + 1) – x = 5

Solution:

2(x + 1) – x = 5

2x + 2 – x = 5 (use distributive property)

x + 2 = 5 (combine like terms)

x + 2 – 2 = 5 – 2

x = 3

Solving Multi-Step Equations With Fractions

To solve an equation with fractions, we first try to change it into an equation without fractions. Then, we can solve it using the methods we already know.

Consider the equation . To remove the fractions, we would need to multiply each term of the equation with the LCM of the denominator. In this case, we will multiply each term with 4. The 4 will cancel and we are left with equation 2x – 12 = 3. To isolate the variable, we will add 12 to both sides and then divide by 2.

Example:

Solve

Solution:

Algebraic Expression - simplify

To simplify an algebraic expression that consists of both like and unlike terms, it might be helpful to first move the like terms together. (When moving the terms, we must remember to move the + or – attached in front of them).

For example,

3x + 2y – 2x + 6

= 3x– 2x + 2y + 6

= (3 – 2)x + 2y + 6

= x + 2y + 6

Example:

Simplify 3x + 2a – 4x

Solution:

3x + 2a – 4x

=3x– 4x + 2a

= (3 – 4)x + 2a

= –x + 2a

Example 1

Example 2:



Example 3

Algebraic Expression - unlike and like

An algebraic expression consisting of two or more like terms can be simplified by combining like terms.

Like terms are terms that have the same variable part i.e. they only differ in their coefficients.

For example:

2x and –5x are like terms

a and are like terms

6x and 5y are unlike terms

Like terms can be added or subtracted from one another.

Example :

Simplify the expressions:

a) 14x + 5x

b) 5y – 13y

c) p – 3p

Solution:

a) 14x + 5x = (14 + 5)x = 19x

b) 5y – 13y = (5 –13)y = –8y

c) p – 3p = (1 – 3)p = – 2p

Algebraic Expression - Coefficient

Coefficients

The coefficient of a variable is the number that is placed in front of a variable.

For example, 3 × w can be written as 3w and 3 is the coefficient.

Coefficient

Coefficients Of Algebraic Terms

The number (positive or negative) in the algebraic term is called the coefficient.

For example:

For the term 4x, 4 is the coefficient

For the term –7y, –7 is the coefficient

The coefficient of 1 in an algebraic term is usually not written.

For example:

1y can be written as simply y. So, 1y and y are the same.

Example:

Find the coefficient and variable of each of the following terms:

a) 0.2m

b) –3p

c) q

d) z

e)

Solution:

a) 0.2m (The coefficient is 0.2 and the variable is m)

b) –3p (The coefficient is –3 and the variable is p)

c) q (The coefficient is and the variable is q)

d) z (The coefficient is 1 and the variable is z)

e) (The coefficient is and the variable is x)

Algebraic Expression - Term

Terms

A term can be any of the following:

• a constant: e.g. 3, 10, π,
• the product of a number (coefficient) and a variable: e.g. –3x, 11y,
• the product of two or more variables: e.g. x², xy, 2y², 7xy

Like terms are terms that differ only in their numerical coefficients. For example: 3a, 22a, are like terms.

Expressions

An expression is made up of one or more terms.

For example:

3w + 4xy + 5

An expression can be term or a collection of terms separated by addition or subtraction operators. Some examples of expressions, with the numbers of terms, are listed below:

Expression	Number of Terms	Description
6x	1	A number multiplied by a variable. The number is always written first followed by the variable(s).
3w – 8	2	Terms separated by –
7b + 5t – 6	3	Terms separated by + and –
	1	All multiplication and division, no + or – symbol

Example:

Determine the number of terms in the following expressions:

a) 5xyz

b) 3x + 2y – 2x + 6

Solution:

a) 5xyz has one term

b) 3x + 2y – 2x + 6 has four terms

Algebraic Expression

In mathematics, often the value of a certain number may be unknown. A variable is a symbol, usually a letter, which is used to represent an unknown number.

Some examples of variables are:

x, a, t, y, b

A term can be a number, a variable, or a number and variable combined by multiplication or division.

Some examples of terms are:

x, 8, 4y,