Combine Like Terms
Multiplication and Division of Terms
Removal of Brackets - Distributive Property
Cross Multiplication
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 termsLike terms can be added or subtracted from one another.
For example:
a + a = 2 × a = 2a (We usually write 2 × a as 2a)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 = 3xyAnswer: 3xy = 1
Example 2:
Simplify: 7a + 5b – 6b + 8a + 2b = 0
Solution:
Step 1: Group together the like terms:
7a + 5b – 6b + 8a + 2b = 0Step 2: Then simplify:
15a + b = 0
Answer: 15a + b = 0
How to Solve an Equation by Combining Like 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
Solving Equations using Multiplication or DivisionSometimes 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
How to Solve an Equation by the Distributive Property?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
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