Download Notes for Solving Linear Equations and Justifying Steps

Notes for Solving Linear Equations and Justifying Steps
Standards: A.REI.3; A.REI.1
Learning Targets:
1. Solve linear equations that have one variable in them.
2. Explain each step in solving simple equations. Use properties to justify the
steps used.
Lesson 1
Properties:
The Commutative Property of Addition states that you can add numbers in
any order.
The Commutative Property of Multiplication states that you can multiply
numbers in any order.
The Associative Property of Addition states that when you are adding you
can group any of the numbers together.
The Associative Property of Multiplication states that when you are
multiplying you can group any of the numbers together.
The Distributive Property states that you can multiply a number by a sum or
multiply be each number in the sum and then add. The result is the same.
Example 1: Simplify the following expressions by combining like terms.
a. 72p – 25p
b.

3 4
x + x4
4
c. 0.5m + 2.5n
d. 3m2 + 5m3
e. 5t3 + t3
Example 2: Simplify 14x + 4(2 + x) and justify each step.
Procedure
Justification
Example 3: Simplify -12x – 5x + 3a + x and justify each step.
Procedure
Justification
Example 4: Simplify 6(x – 4) + 9 and justify each step.
Procedure
Justification
Example 5: Given the equations B = x + 2 and y = A – 4, which of the following
expressions is equivalent to B + A written in terms of x and y?
a. x + y + 2
b. x + y + 6
c. x – y – 4
d. 8xy
e. 2x + 4y
Lesson 2
Relevant Vocabulary:
An _________________________ is a mathematical statement that two
expressions are equal.
A __________________________________________________ is a value of the
variable that makes the equation true.
Properties:
The Addition Property of Equality states that you can add the same number
to both sides of an equation, and the statement will still be true.
The Subtraction Property of Equality states that you can subtract the same
number from both sides of an equation, and the statement will still be true.
The Multiplication Property of Equality states that you can multiply both
sides of an equation by the same number, and the statement will still be true.
The Division Property of Equality states that you can divide both sides of an
equation by the same nonzero number, and the statement will still be true.
*Use inverse operations to solve equations.
Operation
Inverse Operation
Example 1: Solve each equation. Justify each step.
a. y – 8 = 24
b. -6 = k – 6
c. m + 17 = 33
d. 4.2 = t + 1.8
Example 2: Solve each equation. Justify each step.
a. -8 =
j
3

b.
p
= 10
5

c. 9y = 108
d. 16 = -4c
e.
5
w = 20
6

f. 

1 1
 b
4 5
Lesson 3
Always use inverse order of operations when solving two-step and multi-step
equations!!
Example 1: Solve the following equations and justify each step.
a. 18 = 4a + 10
c. –4 +
1
x=3
7

e.
b. 5t – 2 = –32
d.
n
+2=2
7

10y – (4y + 8) = –20
f.

3
1 7
u 
4
2 8
Lesson 4
Relevant Vocabulary:
An _________________________ is an equation that is true for all values of the
variable. This type of equation has infinitely many solutions.
A _________________________ is an equation that is not true for any value of the
variable. It has no solutions.
SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES
To solve an equation with variables on both sides, use inverse operations to
"collect" variable terms on one side of the equation.
HINT: Equations are often easier to solve when the variable has a positive
coefficient. Keep this in mind when deciding on which side to "collect"
variable terms.
Example 1: Solve each equation.
a. 7n – 2 = 5n + 6
b. 4b + 2 = 3b
c. 10 – 5x + 1 = 7x + 11 – 12x
d. 3x + 15 – 9 = 2(x + 2)
e.
1
3
(b  6)  b 1
2
2
f. 4 – 6a + 4a = –1 – 5(7 – 2a)

g. 12x – 3 + x = 5x – 4 + 8x
h. 4y + 7 – y = 10 + 3y
Lesson 5
For the examples below, create an equation and then use it to answer
questions.
Example 1: Over 20 years, the population of a town decreased by 275 people
to a population of 850. Write and solve an equation to find the original
population.
1
of the money he earns from mowing lawns into a
4
college education fund. This year Ciro added $285 to his college education
fund. Write and solve an equation to find how much money Ciro earned
mowing lawns this year.

Example 2: Ciro puts
Example 3: Six times a number plus 5 is equal to 4 less than 3 times the same
number. What is the number?
Example 4: Jon and Sara are planting tulip bulbs. Jon has planted 60 bulbs and
is planting at a rate of 44 bulbs per hour. Sara has planted 96 bulbs and is
planting at a rate of 32 bulbs per hour. In how many hours will Jon and Sara
have planted the same number of bulbs?
Example 5: Four times Greg's age, decreased by 3 is equal to 3 times Greg's
age increased by 7. How old is Greg?
Example 6: The sum of two consecutive whole numbers is 89. What are the
two numbers?
Example 7: Alex belongs to a music club. In this club, students can buy a
student discount card for $19.95. This card allows them to buy CDs for $3.95
each. After one year, Alex has spent $63.40. Write and solve and equation to
find how many CDs Alex bought during the year.