Simplifying Expressions
Simplifying Expressions
MAT 221
Simplifying Expressions
2a(a – 5) + 4(a – 5)
2a(a – 5) + 4(a – 5)
Given
2a(a) – 2a(5) + 4(a) + 4(5)
Left Distributive Property of
Addition Over
Multiplication
2(aa) – 2a(5) + 4(a) + 4(5)
Associative Property of
Multiplication
2(aa) – 2(5)a + 4(a) + 4(5)
Commutative Property of
Multiplication
2(aa) – (2*5)a + 4(a) + 4(5)
Associative Property of
Multiplication
2a2 – 10a + 4a + 20
Multiplication Property
2a2 – 6a + 20
Subtraction Property
On the left side is the step-by-step of the mathematical reasoning for 2a(a – 5) + 4(a – 5)
to be simplified as 2a2 – 6a + 20. On the right side is the step-by-step logical reasoning.
The middle part is combined because the terms are like while the extreme terms are
unlike each of the other three terms. The parentheses are used to show associative
property and then removed via multiplication and subtraction. The numerical coefficient
must be before the literal coefficient.
2w – 3 + 3(w – 4) – 5(w – 6)
Simplifying Expressions
2w – 3 + 3(w – 4) – 5(w – 6)
Given
2w – 3 + 3w + 3(-4) + (-5)w + (-5)(-6)
Left Distributive Property of
Addition Over
Multiplication
2w + 3w – 3 + 3(-4) + (-5)w + (-5)(-6)
Commutative Property of Addition
2w + 3w – 3 + (-5)w + 3(-4) + (-5)(-6)
Commutative Property of Addition
2w + 3w + (-5)w – 3 + 3(-4) + (-5)(-6)
Commutative Property of Addition
2w + 3w – 5w – 3 – 12 + 30
Multiplication Property
5w – 5w – 3 – 12 + 30
Addition Property
0 – 3 + 18
Subtraction Property
0 + 15
Subtraction Property
15
The Additive Identity Is 0
On the left side is the step-by-step of the mathematical reasoning for 2w – 3 + 3(w – 4) –
5(w – 6) to be simplified as 15. On the right side is the step-by-step logical reasoning.
The like terms are combined based on its association with and without the variable w.
The parentheses are used to show associative property and then removed via
multiplication, addition, and subtraction. The numerical coefficient must be before the
literal coefficient. It leads to a positive term.
0.05(0.3m + 35n) – 0.8(-0.09n – 22m)
Simplifying Expressions
0.05(0.3m + 35n) – 0.8(-0.09n – 22m)
Given
0.05(0.3m) + 0.05(35n) + (-0.8)(-0.09n) – 0.8(-22m)
Left Distributive Property of
Addition Over
Multiplication
(0.05*0.3)m + (0.05*35)n + (-0.8*-0.09)n + (-0.8*-22)m
Associative Property of
Multiplication
0.015m + 1.75n + 0.072n + 17.6m
Multiplication Property
0.015m + 1.75n + 17.6m + 0.072n
Commutative Property of
Addition
0.015m + 17.6m + 1.75n + 0.072n
Commutative Property of
Addition
17.615m + 18.22n
Addition Property
On the left side is the step-by-step of the mathematical reasoning for 0.05(0.3m + 35n) –
0.8(-0.09n – 22m) to be simplified as 17.615m + 18.22n. On the right side is the step-bystep logical reasoning. The like terms are combined based on its association with the
variables m and n but cannot be combined together as a one single term. The parentheses
are used to show associative property and then removed via multiplication and addition.
Notice the double negatives. It leads to a positive term. The numerical coefficient must
be before the literal coefficient.
Knowing the properties of real numbers helps me to understand the process of how I go
from one step to the next. It helps me to deduce the facts to the point where further
Simplifying Expressions
simplification is not possible anymore. If I have the axioms and the properties
understood, they must be correctly applied. After checking the word, even through the
rote process, I am liable to be prone to errors in my work. That’s why it helps to go back
to the basics and check my reasoning.