Recall that to simplify an expression there is a need to combine all like terms together through addition and/or subtraction. Algebraic terms are called like terms if they have exactly the same combination of variables.

Just like simplifying algebraic expressions with integer coeffients, we will simplify expressions with rational coefficients.

Ideas

Simplifying expressions with rational coefficients

Simplifying expressions with rational coefficients

The basic rule for simplifying expressions is to combine like terms together and write unlike terms as they are.

To add two or more like terms, add their coefficients and write the common variable with it.

When simplifying expressions with fractions, we have to make sure that the fractions should be in the simplest form and only unlike terms should be present in the simplified expression. For example, \dfrac{3}{9}x+ \dfrac{2}{4}y is not the simplified expression, as fractions are not reduced to their lowest form. On the other hand, \dfrac{x}{3}+\dfrac{y}{2} is in a simplified form as fractions are in the reduced form and both are unlike terms.

For dissimilar fractions or fractions with different denominators, get the least common multiple of the denominators. Rename the fractions by finding the equivalent fractions before adding them.

Examples

Example 1

Simplify the following expression: 0.3 a + 2.5 b - 4.1 a -3.6ab

Worked Solution

Create a strategy

To simplify an expression we combine all the like terms.

Apply the idea

Let's rearrange the expression and group the like terms together so we can clearly see which terms we need to sum.

\displaystyle 0.3 a + 2.5 b - 4.1 a -3.6ab	\displaystyle =	\displaystyle 0.3 a - 4.1 a + 2.5b -3.6ab	Rearrange the expression
	\displaystyle =	\displaystyle (0.3 a - 4.1 a) + 2.5 b -3.6ab	Group the like terms
	\displaystyle =	\displaystyle -3.8a + 2.5b - 3.6ab	Simplify

Reflect and check

We identified like terms and then combined them until no like terms remained. We can add any of the terms together regardless of the ordering of the expression.

Example 2

Simplify \dfrac{3}{8} x + \dfrac{5}{6} y - \dfrac{1}{2} y - \dfrac{1}{4} x.

Worked Solution

Create a strategy

When adding or subtracting expressions with fractions of different denominators, get the least common multiple of the denominator of the expressions with like terms before combining them.

Apply the idea

Let's rearrange the expresions, group the like terms, find the LCM of the denominator and add or subtract.

\displaystyle \dfrac{3}{8} x + \dfrac{5}{6} y - \dfrac{1}{2} y - \dfrac{1}{4} x	\displaystyle =	\displaystyle \dfrac{3}{8} x - \dfrac{1}{4} x+ \dfrac{5}{6} y - \dfrac{1}{2} y	Rearrange the expression
	\displaystyle =	\displaystyle \left(\dfrac{3}{8} x - \dfrac{1}{4} x\right)+ \left( \dfrac{5}{6} y - \dfrac{1}{2} y\right)	Group the like terms
	\displaystyle =	\displaystyle \left(\dfrac{3}{8} x - \dfrac{2}{8} x\right)+ \left( \dfrac{5}{6} y - \dfrac{3}{6} y\right)	Rename the fractions to have the same denominators
	\displaystyle =	\displaystyle \dfrac{1}{8} x - \dfrac{2}{6} y	Evaluate
	\displaystyle =	\displaystyle \dfrac{1}{8} x - \dfrac{1}{3} y	Simplify

Reflect and check

We can also rewrite the expression \dfrac{1}{8} x - \dfrac{1}{3} y as \dfrac{x}{8} - \dfrac{y}{3}.

Idea summary

In simplifying expressions with rational exponents:

Combine like terms together and write unlike terms as they are.
To add or subtract like terms, add or subtract their coefficients and then write the common variable with it.
For expressions with fractions, combine like terms and find the least common denominator of dissimilar fractions.
Simplified expressions with fractions should be in simplest or lowest terms.

Outcomes

7.EE.A.1

Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

5.02 Simplify expressions with rational coefficients

Introduction