Download Simplifying with variables

Simplifying with variables
Variables cont’d
We simplify variables similar to numbers
meaning:
Ex. 2x + 3x
5x
We add the numbers 2 and 3 because they
both have an “x” and put the “x” on the end.
Examples
• 4x + 2x
6x
What about this one?
5m + 2m + m – 10m
7m + m – 10m
8m – 10m
-2m
12m – 17m
-5m
2 variables
Ex. 2s + 3r – 4s + 6r
With these we arrange the like variables
together and simplify
***Make sure to keep the sign in front of them****
2s – 4s + 3r + 6r
-2s + 9r
Do we go any further?
NO
You do
• 5x – 13x + 2x
6x – 4y + x – 7w
• 10s – 20r + 4r – 6s
14z + 5y - 9z+10y
Mathematical Properties
• Associative Property –
– a + (b + c) = (a + b) + c or (ab)c – a(bc)
• Commutative
– s+7+r=7+s+r
or t*r*s = s*t*r
• Identity
– 16 + 0 = 16 or r * 1 = r
Mathematical Prop. Con’t
• Substitution– Ex. 6+ 4 = 10 or 8 – 5 = 3 or 7(3) = 21
• Distributive– a(b + c) = ab * ac
– s(r – t + v) = sr – st + sv
Prerequisite Skills
SKILL CHECK
Solve the equation using mental math.
5.
x –1 = 5
ANSWER
6
6.
x –2 = 9
ANSWER
11
7.
4 + x = 12
ANSWER
8
8.
10 = x + 3
ANSWER
7
9.
5x = 20
ANSWER
4
Prerequisite Skills
SKILL CHECK
10.
x
3=1
ANSWER
3
11.
x
7=5
ANSWER
35
12.
3x = 51
ANSWER
17
Solving equations
• Remember “PEMDAS” in simplifying we
do the opposite “SADMEP” when solving.
• ***Following this process you will never
get a problem wrong.***
SADMEP
• SADMEP means do the opposite of what
you see in that order, to the variable.
• Subtraction – going to add
• Addition – going to subtract
• Division – going to multiply
• Multiplication – going to divide
• Exponents – going to take the root
• Parentheses – do what’s inside last
Solving an Equation Using Subtraction
EXAMPLE 1
x+ 8
= –15
x+ 8
= –15
–8
x
–8
= –23
ANSWER
The solution is –23.
Original equation
Subtract 8 from each side to undo addition.
Simplify. x is by itself.
EXAMPLE 2
Solving an Equation Using Addition
c – 4.5 = 13
c – 4.5 = 13
+ 4.5 = + 4.5
c = 17.5
Check
17.5 – 4.5
=? 13
13 = 13 ✓
Add 4.5 to each side to undo
subtraction.
Simplify. c is by itself.
Substitute 17.5 for c
in original equation.
EXAMPLE 3
Using a Model
Rock Climbing
A cliff has a height of about 1500 feet. If you
have already climbed 675 feet, how much
farther do you have to climb to reach the top?
SOLUTION
Use the diagram to help you write an
algebraic model. Let x represent the
distance left to climb.
EXAMPLE 3
Using a Model
1500 = x + 675
1500 – 675 = x + 675 – 675
825 = x
ANSWER
You have about 825 feet left to climb.
Write an algebraic model.
Subtract 675 from each side.
Simplify. x is by itself.
for Examples 1, 2, and 3
GUIDED PRACTICE
Solve the equation. Check your solution.
1.
x + 9 = 20
x + 9 = 20
c + 9 – 9 = 20 – 9
x = 11
Check
11 + 9 =? 13
11 = 11 ✓
Original equation
Subtract 9 from each side.
Simplify. x is by itself.
Substitute 11 for x
in original equation.
for Examples 1, 2, and 3
GUIDED PRACTICE
2. –10 = 3 + y
–10 = 3 + y
–3 –3
Original equation
Subtract 3 from each side.
–13 = y
Check
Simplify. y is by itself.
Substitute –13 for y
–10 =? 3 – 13
–10 = –10 ✓
in original equation.
for Examples 1, 2, and 3
GUIDED PRACTICE
3.
m – 14 = –15
m – 14 = –15
+ 14
+14
m = –1
Check
–1 – 14 =? –15
–1 = – 1 ✓
Original equation
Add 14 from each side.
Simplify. m is by itself.
Substitute –1 for m
in original equation.
for Examples 1, 2, and 3
GUIDED PRACTICE
4.
2 = z – 6.4
2 = z – 6.4
2 + 6.4 = z – 6.4 + 6.4
8.4 = z
Check
Original equation
Subtract 6.4 from each side.
Simplify. z is by itself.
2 =? 8.4 – 6.4
Substitute 8.4 for z
2=2✓
in original equation.
GUIDED PRACTICE
5.
for Examples 1, 2, and 3
Seashells
Lucinda combines her 49 seashells with Jerry’s seashells, for a total of
162. Write and solve an addition equation to find how many seashells
Jerry had before their collections were combined.
SOLUTION
Let s represent Jerry’s seashells
s + 49 = 162
s + 49 – 49 = 162 – 49
s = 113
ANSWER
Write an algebraic model.
Subtract 49 from each side.
Simplify. s is by itself.
Jerry had 113 seashells before their collections
were combined
Examples for Multiplying
2x = 8
**Remember a letter and number together
is multiplication.
2x = 8
2x = 8
2
2
x=4
We see multiply so divide
both sides by 2
The 2’s cancel and 8
divided by 2 is 4
More examples
3x = 18
3x = 18
3
3
x = 6
-5x = 20
-5x = 20
-5 -5
x = -4
•
•
•
•
4x = -12
10x = 90
-8x = 48
-3x = -21
Examples Dividing
• We keep our same rules of SADMEP
• So b/c we see divide we multiply both
sides as the review we did earlier.
y
3
=5
1. 119 = 31 + h
2. 59 = 65 - a
3. e ÷ 3 = 2
• 4. 9g = 27
5. u - 11 = 9
6. 21 + w = 30
7. 9 = x ÷ 8
8. 62 = z + 54
9. m + 25 = 115
10. 36 = j - 32
11. 62 = 93 - d
12. 36 ÷ f = 6
13. 139 = b + 98
14. q + 71 = 136
•