Lesson 2.1 Solving Equations
w/Justification
Concept: Solving Equations
EQ: How do we justify how we solve equations? REI. 1
Vocabulary:
Properties of Equality
Properties of Operation
Justify
1
Solve the equations below, provide an
explanation for your steps.
1.
2x – 3 = 13
2.
3𝑥+1
2
=5
2
Properties of Equality
Property
In symbols
Example
Reflexive property
a=a
of equality
2=2
Symmetric
property
of equality
If a = b, then
b = a.
x=3
3=x
Transitive
property
of equality
If a = b and b = c,
then a = c.
x = 2, y = 2, x = y
Addition property
of equality
If a = b, then a +
c = b + c.
x–4=3
x–4+4=3+4
x=7
3
2.1.1: Properties of Equality
Properties of Equality, continued
Property
In symbols
Subtraction
If a = b, then
property of equality a – c = b – c.
If a = b and
Multiplication
c ≠ 0, then
property of equality
a • c = b • c.
Division property
of equality
If a = b and
c ≠ 0, then
a ÷ c = b ÷ c.
Examples
x + 2 =5
x+2–2=5–2
x=3
x=15
4x = 16
x=4
4
2.1.1: Properties of Equality
Properties of Equality, continued
Property
In symbols
Examples
If a = b, then b
may be
Substitution
substituted for
property of equality a in any
expression
containing a.
x = 3, then
2x = 2(3) = 6
5
2.1.1: Properties of Equality
Properties of Operations
Property
General rule
Commutative property of
a+b=b+a
addition
Associative property of
addition
Specific example
3+8=8+3
(a + b) + c = a + (b + c) (3 + 8) + 2 = 3 + (8 + 2)
Commutative property of
a•b=b•a
multiplication
3•8=8•3
Associative property of
multiplication
(a • b) • c = a • (b • c)
(3 • 8) • 2 = 3 • (8 • 2)
Distributive property of
multiplication over
addition
a • (b + c) = a • b + a • c 3 • (8 + 2) = 3 • 8 + 3 • 2
6
2.1.1: Properties of Equality
Guided Practice
Example 1
Which property of equality is missing in the steps
to solve the equation –7x + 22 = 50?
Equation
–7x + 22 = 50
Steps
Original equation
–7x = 28
x = –4
Division property of equality
7
2.1.1: Properties of Equality
Guided Practice: Example 1, continued
1. Observe the differences between the
original equation and the next equation in
the sequence. What has changed?
Notice that 22 has been taken away from both
expressions, –7x + 22 and 50.
8
2.1.1: Properties of Equality
Guided Practice: Example 1, continued
2. Refer to the table of Properties of
Equality.
The subtraction property of equality tells us that when
we subtract a number from both sides of the equation,
the expressions remain equal.
The missing step is “Subtraction property of equality.”
✔
9
2.1.1: Properties of Equality
Guided Practice: Example 1, continued
10
2.1.1: Properties of Equality
Guided Practice
Example 2
Which property of equality is missing in the steps to
solve the equation −3
Equation
𝑥
−
6
= 4?
Steps
Original equation
−𝑥
=7
6
Addition property of equality
–x = 42
x = –42
Division property of equality
11
2.1.1: Properties of Equality
Guided Practice: Example 2, continued
1. Observe the differences between the
original equation and the next equation in
the sequence. What has changed?
Notice that 3 has been added to both expressions,
−3
𝑥
−
6
and 4. The result of this step is
𝑥
−
6
= 7.
12
2.1.1: Properties of Equality
Guided Practice: Example 2, continued
In order to move to the next step, the division of 6 has
been undone.
The inverse operation of the division of 6 is the
multiplication of 6.
The result of multiplying -
x
by 6 is –x and the result
6
of multiplying 7 by 6 is 42. This matches the next step
in the sequence.
13
2.1.1: Properties of Equality
Guided Practice: Example 2, continued
2. Refer to the table of Properties of
Equality.
The multiplication property of equality tells us that
when we multiply both sides of the equation by a
number, the expressions remain equal.
The missing step is “Multiplication property of
equality.”
✔
14
2.1.1: Properties of Equality
Guided Practice: Example 2, continued
15
2.1.1: Properties of Equality
Guided Practice: Example 3
What equation is missing based on the steps?
1. Observe the 3rd and 5th equations.
2. Read the 4th step.
3. Fill in the missing equation.
2.1.1: Properties of Equality
16
You Try…
Identify the property of equality that justifies each
missing step or equation.
Equation
Steps
3.
9 + x = 17
Original Equation
x=8
Equation
7(2x + 1) = 49
4.
Steps
Original Equation
14x + 7 = 49
14x = 42
x=3
Subtraction Property of
Equality
17
5. Solve the equation that follows. Justify each step in
your process using the properties of equality. Be sure to
include the properties of operations, if used.
8(2x – 1) = 56
18
Summary…
Identify the property represented below.
1. x -3 = 6
x-3+3=6+3
2. A = B, B = C, then A = C
Solve the problem below justifying each step using the
properties of equality.
3. 2x – 9 = 1
19
Solving Equations with the Variable in
Both Expressions of the Equation
1. Move the variable to solve for to the left of the
2.
3.
4.
5.
equal sign.
Move all other terms to the right of the equal
sign.
Combine like terms on each side of the equal
sign.
Now solve for the variable and simplify.
Substitute the solution into the original
equation and check your work.
20
Example 4: Solve the equation
5𝑥 + 9 = 2𝑥 − 36
21