Like Terms, Coefficients and Simplification of Algebraic Expressions for JSS 1

Simplification of Algebraic Expressions

Performance Objectives

By the end of the lesson, students should be able to:

1. identify and collect like and unlike terms of given algebraic expressions;
2. identify coefficients of given algebraic terms;
3. identify the positive and negative coefficients of given algebraic terms;
4. perform basic arithmetic operations on algebraic expressions of similar terms;
5. solve word problems involving algebraic expressions;
6. inserting and removing of brackets in simplifying algebraic expressions;
7. solve quantitative aptitude problems involving algebraic expressions.

Introduction to Algebraic Expressions

An algebraic expression is a mathematical phrase that contains numbers, variables and operation signs.

A variable is a letter or symbol that represents an unknown number.

A constant is a fixed number that does not change its value.

A term is a single number, a variable, or numbers and variables multiplied together.

In the algebraic expression $4x + 7$, the letter $x$ is the variable, the number $4$ is multiplied by the variable, and $7$ is the constant. The expression consists of two terms, which are $4x$ and $7$. These terms are separated by the addition sign.

Identifying Like and Unlike Terms

Like terms are algebraic terms that have the exact same variables raised to the exact same powers.

Unlike terms are algebraic terms that have different variables, or the same variables raised to different powers.

Only the numerical parts of like terms can be different. For example, $3a$ and $7a$ are like terms because they both share the variable $a$. However, $3a$ and $3b$ are unlike terms because their variables are different. Similarly, $5x$ and $5x^2$ are unlike terms because the variable $x$ is raised to different powers in each term.

Examples of groups of like terms include:

1. $2x$, $5x$, $-9x$ and $x$
2. $4ab$, $-3ab$, $12ab$ and $ba$
3. $7y^2$, $2y^2$ and $-y^2$

Examples of groups of unlike terms include:

1. $3x$ and $3y$ (different variables)
2. $5a$ and $5a^2$ (different powers of the same variable)
3. $8pq$ and $2p$ (different variable combinations)

Observe the diagram below to see how like and unlike terms are grouped.

Identifying Coefficients

A coefficient is the numerical factor of an algebraic term that multiplies the variable.

The coefficient includes the positive or negative sign that comes before the number. If no number is written before a variable, the coefficient is understood to be $1$. If there is only a negative sign before the variable, the coefficient is $-1$.

The table below shows examples of algebraic terms, their coefficients and the type of coefficient:

Basic Arithmetic Operations on Algebraic Expressions

Basic arithmetic operations such as addition and subtraction can only be performed on like terms. When adding or subtracting like terms, the coefficients are added or subtracted while the variable remains unchanged. Unlike terms cannot be added or subtracted to form a single term.

Worked Example 1

Simplify $8x + 5x$.

Solution:

$$
\begin{aligned}
8x + 5x &= (8 + 5)x \\
&= 13x
\end{aligned}
$$

Worked Example 2

Simplify $12y – 4y$.

Solution:

$$
\begin{aligned}
12y – 4y &= (12 – 4)y \\
&= 8y
\end{aligned}
$$

Worked Example 3

Simplify $-3a – 7a$.

Solution:

$$
\begin{aligned}
-3a – 7a &= (-3 – 7)a \\
&= -10a
\end{aligned}
$$

Simplifying Algebraic Expressions by Grouping Like Terms

When an algebraic expression contains both like and unlike terms, simplification is achieved by collecting the like terms together before performing the arithmetic operations. It is important to move each term along with the positive or negative sign that precedes it.

Worked Example 1

Simplify $9a + 5b – 4a + 2b$.

Solution:

First, collect the like terms together:

$$
\begin{aligned}
9a + 5b – 4a + 2b &= 9a – 4a + 5b + 2b \\
&= (9 – 4)a + (5 + 2)b \\
&= 5a + 7b
\end{aligned}
$$

Worked Example 2

Simplify $12x – 8 – 5x + 11$.

Solution:

First, collect the like terms (the $x$ terms together and the constant numbers together):

$$
\begin{aligned}
12x – 8 – 5x + 11 &= 12x – 5x – 8 + 11 \\
&= (12 – 5)x + (-8 + 11) \\
&= 7x + 3
\end{aligned}
$$

Worked Example 3

Simplify $3p – 4q – 8p + 9q$.

Solution:

First, collect the like terms together:

$$
\begin{aligned}
3p – 4q – 8p + 9q &= 3p – 8p – 4q + 9q \\
&= (3 – 8)p + (-4 + 9)q \\
&= -5p + 5q
\end{aligned}
$$

Using Brackets in Algebraic Expressions

Brackets are used in algebra to group terms together. Simplifying expressions with brackets involves either removing the brackets (expansion) or inserting brackets (factorisation).

Removing Brackets

To remove brackets, multiply the term outside the bracket by every term inside the bracket. This is known as the distributive law:

$$a(b + c) = ab + ac$$

Special attention must be paid to negative signs outside brackets. A negative sign outside a bracket changes the sign of every term inside the bracket when the bracket is removed:

$$-(x + y) = -x – y$$
$$-(x – y) = -x + y$$

Worked Example 1

Expand and simplify $3(2x + 5) + 4x$.

Solution:

First, multiply $3$ by each term inside the bracket:

$$
\begin{aligned}
3(2x + 5) + 4x &= 6x + 15 + 4x \\
&= 6x + 4x + 15 \\
&= 10x + 15
\end{aligned}
$$

Worked Example 2

Expand and simplify $5(2a – 3b) – 2(3a – 4b)$.

Solution:

Multiply the terms, paying close attention to the negative sign before the second bracket:

$$
\begin{aligned}
5(2a – 3b) – 2(3a – 4b) &= 10a – 15b – 6a + 8b \\
&= 10a – 6a – 15b + 8b \\
&= 4a – 7b
\end{aligned}
$$

Inserting Brackets

To insert brackets, find the highest common factor (HCF) of the terms and place it outside the bracket. Then, divide each term by this factor to find the terms inside the bracket.

Worked Example 1

Insert brackets in the expression $4x + 12$.

Solution:

The highest common factor of $4x$ and $12$ is $4$. Divide each term by $4$:

$$
\begin{aligned}
\frac{4x}{4} &= x \\
\frac{12}{4} &= 3
\end{aligned}
$$

Write the common factor outside the bracket:

$$4x + 12 = 4(x + 3)$$

Worked Example 2

Insert brackets in the expression $6ab – 9ac$.

Solution:

The highest common factor of $6ab$ and $9ac$ is $3a$. Divide each term by $3a$:

$$
\begin{aligned}
\frac{6ab}{3a} &= 2b \\
\frac{-9ac}{3a} &= -3c
\end{aligned}
$$

Write the common factor outside the bracket:

$$6ab – 9ac = 3a(2b – 3c)$$

Word Problems Involving Algebraic Expressions

Word problems can be solved by translating English statements into algebraic expressions and then simplifying them. Letters are used to represent the unknown quantities.

Worked Example 1

Amina has $y$ exercise books. Chidi has $5$ more exercise books than Amina. Babajide has twice as many exercise books as Amina. Write an expression for the total number of exercise books they have altogether and simplify it.

Solution:

Represent each person’s books algebraically:

1. Amina’s books = $y$
2. Chidi’s books = $y + 5$
3. Babajide’s books = $2y$

Add the expressions together to find the total:

$$
\begin{aligned}
\text{Total books} &= y + (y + 5) + 2y \\
&= y + y + 2y + 5 \\
&= 4y + 5
\end{aligned}
$$

Worked Example 2

A farmer harvested $10x$ tubers of yam. He sold $4x$ tubers on Monday and $3x$ tubers on Tuesday. Write down and simplify an expression for the number of yam tubers left.

Solution:

Represent the total yams sold and subtract from the total harvested:

$$
\begin{aligned}
\text{Yams left} &= 10x – (4x + 3x) \\
&= 10x – 7x \\
&= 3x
\end{aligned}
$$

Quantitative Aptitude Problems

Quantitative aptitude problems involve finding mathematical patterns or rules in diagrams and applying them to solve algebraic expressions.

Worked Example 1

Consider a triangular pattern where the expression in the top vertex is the sum of the expressions in the two bottom vertices. If the bottom vertices contain $3x + 2y$ and $2x – 5y$, find the expression for the top vertex.

Solution:

Add the two bottom expressions together:

$$
\begin{aligned}
\text{Top Vertex} &= (3x + 2y) + (2x – 5y) \\
&= 3x + 2x + 2y – 5y \\
&= 5x – 3y
\end{aligned}
$$

Look at the pattern diagram below to see how this relationship works.

Worked Example 2

In a rectangular grid, the expression in the right box is obtained by subtracting the left box expression from the middle box expression. If the middle box is $8a – 3b$ and the left box is $3a – 5b$, find the expression for the right box.

Solution:

Subtract the left box expression from the middle box expression:

$$
\begin{aligned}
\text{Right Box} &= (8a – 3b) – (3a – 5b) \\
&= 8a – 3b – 3a + 5b \\
&= 8a – 3a – 3b + 5b \\
&= 5a + 2b
\end{aligned}
$$

Practice Exercises

Solve the following algebraic problems:

1. Identify the like terms in the following list: $4x$, $3y$, $-5x$, $7x^2$, $x$, $12y$.
2. State the coefficient of $y$ in each of the following terms:
   1. $-8y$
   2. $y$
   3. $\frac{3}{4}y$
   4. $-y$
3. Simplify the expression: $7a + 8a – 3a$.
4. Simplify the expression: $10x – 4y – 3x + 9y$.
5. Simplify the expression: $-5p + 12q – 2p – 7q$.
6. Remove brackets and simplify: $4(3x + 2) – 5x$.
7. Remove brackets and simplify: $3(2a – 5b) – 2(a – 3b)$.
8. Insert brackets in the following expressions by finding the highest common factor:
   1. $5x + 15$
   2. $8ab – 12ac$
9. Ngozi has $p$ oranges. Olu has $4$ fewer oranges than Ngozi. Emeka has $3$ times as many oranges as Ngozi. Write down and simplify an expression for the total number of oranges they have altogether.
10. In a quantitative aptitude pattern, the number in a square is the product of the number outside and the algebraic expression in the bracket: $\text{Square} = 5(2x – 3y)$. Expand this expression to find the value in the square.