Simplification of algebraic expressions

Simplification of algebraic expressions

Definition of algebraic expressions

Algebraic expressions involve the operations of addition, subtraction, multiplication or division. Algebraic expressions are simplified by performing addition or subtraction of like terms. Algebraic expressions with like variables and exponents are called like terms. For example, $8y$ and $19y$, or $8yx$ and $3xy$, or $6x^2y$ and $2x^2y$ are like terms. Algebraic expressions with unlike variables and exponents are called unlike terms. For example, $6x$ and $6z$, or $3mn$ and $4vy$, or $5u^2y$ and $6y^2v$ are unlike terms. Terms can have coefficients which are whole numbers, fractions, decimals or mixed numbers.

Terms with different coefficients

Examples of algebraic expressions having terms with whole numbers are: $3x + 5x + 15y$ and $4m$.

Examples of algebraic expressions having terms with fractions are: $\frac{1}{3}x + \frac{2}{3}x$ and $\frac{1}{2}y + 2$.

Examples of algebraic expressions having terms with decimals are: $0.4x + 0.5x + 1.5y$ and $0.6m - 2.5m$.

Examples of algebraic expressions having terms with mixed numbers are: $4\frac{1}{3}x + \frac{2}{3}x + 0.6y$ and $4x + \frac{2}{5}x + 9y - 0.7$.

Addition and subtraction of algebraic expressions with whole number coefficients

Example 1

Simplify: $12xy + 7xz^2 + 2(5xy - xz^2)$

Solution

Open brackets: $12xy + 7xz^2 + 2(5xy - xz^2) = 12xy + 7xz^2 + 10xy - 2xz^2$

Collect like terms: $12xy + 7xz^2 + 10xy - 2xz^2 = 12xy + 10xy + 7xz^2 - 2xz^2$

Simplify by adding or subtracting like terms: $12xy + 10xy + 7xz^2 - 2xz^2 = 22xy + 5xz^2$

Therefore, $12xy + 7xz^2 + 2(5xy - xz^2) = 22xy + 5xz^2$

Example 2

Simplify: $14y - 6xy + 2y - xy$

Solution

Collect like terms as follows: $14y - 6xy + 2y - xy = 14y + 2y - 6xy - xy$

Simplify by adding or subtracting like terms: $14y + 2y - 6xy - xy = 16y - 7xy$

Therefore, $14y - 6xy + 2y - xy = 16y - 7xy$

Example 3

Simplify: $2y - 3(x + y) + 10x$

Solution

Open brackets: $2y - 3(x + y) + 10x = 2y - 3x - 3y + 10x$

Collect like terms: $2y - 3x - 3y + 10x = 2y - 3y - 3x + 10x$

Simplify by adding or subtracting like terms: $2y - 3y - 3x + 10x = -y + 7x$

Therefore, $2y - 3(x + y) + 10x = 7x - y$