Download U1 L5 Using Properties to Solve Equations

Unit 1: Relationships between Quantities and Reasoning with Equations
Lesson 5- Using Properties to Solve Equations
Objectives:
 I can identify the properties of real numbers and properties
of equality.
 I can use the properties to justify each step of a solution and
solve equations.
Warm Up: State the inverse.
Addition -
Subtraction-
Squaring-
Multiplication-
Division-
Square root-
Properties of Real Numbers:
Whenever you simplify or evaluate an expression, you use properties
of real numbers. You probably use these properties without even
realizing it. They are what justify many of the steps you take when
working with expressions and equations.
Here are some of the important properties of real numbers…
Property
Associative
Addition
Multiplication
a+(b + c) = (a + b) + c a (bc) = (ab)c
Commutative
a+b = b+a
ab = ba
Identity
a+0=a
a1 = a
Inverse
a + (-a) = 0
a  (1/a) = 1
Distributive
a (b + c) = ab + ac
*Note the multiplicative inverse of a number is also called the
reciprocal and the additive inverse of a number is also called the
opposite.
You try!
Identify the property.
3(5x – 1)
5+0=5
3(42) = (34)2


61 = 6
3+2=2+3
8 + (-8) = 0
Properties are also useful for manipulating equations in order to
find solutions. Look at the next page for the properties of
equality.
The properties of equality listed below can be used to isolate
variables and find their values.
Properties of Equality
Addition Property
If a = b, then a+c = b+c.
Subtraction Property
If a = b, then a - c = b - c.
Multiplication Property
If a = b, then ac = bc.
𝑎
𝑏
Division Property
If a = b and c ≠ 0, then = .
𝑐
𝑐
Reflexive Property
a=a
Symmetric Property
If a = b, then b = a.
Transitive Property
If a = b and b = c, then a = c.
Substitution Property
If a = b, then b can replace a in any
expression.
Felicia wrote the steps shown below while solving the equation
below. Use the properties of real numbers and properties of equality
to justify each step in Felicia’s solution.
3(4 + x) = -3.
Step 1: 12 + 3x = -3
____________________
Step 2: 3x + 12 = -3
____________________
Step 3: 3x = -15
____________________
Step 4: x = -5
____________________
Think…
Can you think of a way to check to see if the solution is correct?
How?
Brandon was given the equation 36 = 5x + (3y – 7x) and was asked
to write it in slope-intercept form, y = mx + b. The steps Brandon took
are shown below. Use properties to justify each step in the solution.
36= 5x + (3y – 7x)
Step 1: 36 = 5x + (-7x + 3y) ____________________
Step 2: 36 = [5x + (-7x)] + 3y ____________________
Step 3: 36 = -2x + 3y
____________________
Step 4: 2x + 36 = 3y
____________________
Step 5:
2
3
𝑥 + 12 = 𝑦
Step 6: 𝑦 =
2
3
𝑥 + 12
____________________
____________________
Kevin was given the equation 3x + 5y = 12 and y = 2 – x. He used
the following steps to find the x-value of the coordinate pair that
satisfies both equation.
3x + 5y = 12
Step 1: 3x + 5(2 – x) = 12
_____________________
Step 2: 3x + 10 – 5x = 12
_____________________
Step 3: -2x + 10 = 12
______________________
Step 4: -2x = 2
______________________
Step 5: x = -1
______________________
To find the y-value, Kevin took the following steps.
y=2–x
Step 1: y = 2 – (-1)
Step 2: y = 3
Which property justifies Step 1? ________________________
Solving Equations:
Now that we have looked at how the properties are used to justify the
steps of a solution, let’s solve!
We can follow the following 5 step process when we are solving
equations…
Step 1: Distribute
Step 2: Add like terms on each side of the equation
Step 3: Get the variables on one side of the equation
Step 4: Get rid of the constant by using the additive inverse
Step 5: Get rid of the coefficient by using the multiplicative
inverse
Let’s try a few…
2x – 5 = 7
3(2x + 4) = -12
-x + 4x + 2 = 8
6x + 15 = 3x + 8
3(2x + 5) = 4x + 7 – x + 1
5𝑦 − 2
=3
4
You try!
3x – 5 = 16
3x + 5 – 2x = 6x – 10
6x + 2(2x + 3) = 16
3 – 2(x + 4) = -3(4x – 5)
====================================================
Write your own equation that requires multiple steps
to solve. Solve it and justify each step of the solution.
Name: _________________________
Unit 1 Lesson 5: Using Properties to Solve Equations
Identify the property of real numbers that is demonstrated.
1) 12 + 0 = 12
2) 9 + d = d + 9
3) 7(2 – p) = 14 – 7p
4) 6p = 16p
Identify the property of equality that is demonstrated.
5) 13x7 = 13x7
6) If z = 12 and 12 = 34, then
z = 34.
7) If 12t = 5s, then 5s = 12t.
8) If n = 0.25p and 3p + 2n = 14,
then 3.5p = 14.
Use properties to justify each step taken to solve the equations.
9) 8 + 7x – 8 = 49
8 – 8 + 7x = 49 ______________________
0 + 7x = 49
______________________
7x = 49
______________________
x=7
______________________
Solve for x. Use properties to justify the steps you use.
10) x + 6 = 8
11) y = -5
x=y
Property: ________________
Property: _________________
12) The associative property of multiplication states that
(a x b) x c = a x (b x c). How would you express this property in
words?
13) Alexis solved the equation 5x = 4 in the two ways shown below.
Method 1
5x = 4
5x ÷ 5 = 4÷5
x=
4
5
Method 2
5x = 4
1
5
∙ 5𝑥 = 4 ∙
x=
1
5
4
5
Use properties to justify the steps in each of Alexis’s solution
methods. What do Alexis’s methods tell you about the relationship
between the properties she used?
14) Solve.
2(-x + 4) = 5x – 4
15) Solve.
𝑥+3
=5
2