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One Step Equations – AdditionOne Step Equations – Addition
Two–Step Equations – Multiplication1. Look at the variable side, find the
constant, and get rid of it first.
2. To get rid of ‒7, add the opposite (+7)
+ 7 +7
3. Cancel the opposites...
… bring down the variable term
…then add. 3x =
4. To get rid of the coefficient, 3 ……
… DIVIDE both sides by 3
3 3
x = 2
‒8 ‒8
x = 2
2 2
x =
1. Look at the variable side, find the constant, and get rid of it first.
2. To get rid of 8, add the opposite (‒8) 3. Cancel the opposites...
… drop the variable term…then add.
4. To get rid of x divided by 2, …
… MULTIPLY both sides by 2
a constant is a number without a variable – it’s the “naked
number”
a coefficient is the number in front of the variable
6
‒18
Two–Step Equations – Division
–––– –– –––– ––
––––
++++++++++++++
++++++++++++++
++ ++++++ ++ ++
++++
++++++++
++++++++
–– ––
––
–– –––– ––
–––– ––
–– –––– –––– –––– ––
–– –––– –––– –––– ––
–– ––
––
–– ––––
––––
–– –––– –––– –––– –– ––
–––– ––
––
–– ––––
––––
–– –––– –––– –––– –– ––
––
–– ––––
–– ––––––
–––– –––– –––– –––– –– ––
–––– ––
–– –– ––––
––––
–– –––– –––– –––– –– ––
––‒ 36
Two–Step Equations – Multiplication
+14+14
= 222 2
Two–Step Equations – Division
– 12
– 12
4– = – 2x –2 –2
x2 =
4
x = 442
x
= 16 – a6‒16‒16
‒10 = – aRemember,
‒ a = ‒1aSo, stick a 1 in front of the a.
1
‒10 = – a1
‒1 ‒1
10 = a
9 = ‒ y + 12
7
If you have a
negative sign just sitting in front of a fraction, move it next to
the constant.
9 = y + 12 ‒12 ‒ 12
‒3= ‒7-7
21 = y
x =
‒ 3 = ‒27 + y
8
= y
–3
192
–7
EXAMPLE 2 Negative six, increased by the product of four and a number, is negative twenty-two.
n = –4
Negative six
+
the product of four and a number
–6 4n =
Fifteen is twenty-six less than the quotient of a number and negative three.
Writing and Solving a Two-Step EquationWriting and Solving a Two-Step Equation1.
2.
increased by isnegative twenty-two.
–22+6 +6 4n = –
16 4 = 4
The number is negative
four.
Fifteen15
is
=
twenty-six
26less than
–the quotient of a number and negative three.n
–3 + 26 + 26 41 n_
–3(–3)(–3) =
=–123 n
The number is negative
one hundred
seventeen.
Writing and Solving a Two-Step EquationWriting and Solving a Two-Step EquationYour online music website charges a monthly fee of $8, plus $0.35 for every songsong you download. If you paid $13.25 last month, how many
songssongs did you download?1. Read it again, and pick out the TOTAL.Set a blank equation equal to
13.25 = 13.25
2. Now, figure out HOW you get to that
total.
monthly fee + songs = TOTAL
8 + 0.35x
3. Solve for x (songs).x (songs).
Moe, Larry, and Curley are equal partners in a lemonade stand. To calculate each person’s earnings, they’ll take the total money madetotal money made, divide it by three, then subtract $2 (for supplies). If each stooge got
$43, what was the total money madetotal money made?1. Read it again, and pick out the TOTAL.Set a blank equation equal to
43 2. Now, figure out HOW you get to
that total.3. Solve for x (total money made).x (total money made).
= 43
total money – supplies = TOTAL 3
x – 2 3
x = 15
You downloaded fifteen
songs
x = 135
The total money
made was $135.
Solving Equations by Solving Equations by Combining Like Combining Like TermsTerms3x +12 – 4x =
20Look: There are 2 variable terms …
… so, COMBINE LIKE TERMS first.–1x +12 =
20
Remember,
‒1x = ‒x but, just
leave the 1 there.
– 12 – 12
–1x = 8–1 –1
x =
1. Look at the variable side, find the constant, and get rid of it first.
2. To get rid of +12, add the opposite (‒12)
3. Cancel the opposites …… bring down the variable term
4. To get rid of the coefficient, ‒1 … …
… DIVIDE both sides by ‒1
…then add.
–8
w = – 1
–6 = 11w –5w1.
Solve the equation.
p = 3
2. 4p +10 + p = 25
r = 7
3. –8r – 2 + 7r = – 9
Solving Equations by Solving Equations by Combining Like Combining Like TermsTerms
EXAMPLE 3
6n –2(n +1) = 26
Use Distributive property
Combine like terms.
4n = 28Solve.
n = 7
Add 22 to each side.
6n –2(n +1) = 26
““outer times first”, outer times first”, then
––2n2n
““outer times second”, outer times second”,
––226n = 26
4n 4n – 2 = 26 + 2 + 2
Solving Equations by using Solving Equations by using Distributive Distributive PropertyProperty
2
3x = or – 4x = – 4
3(x – 9) = – 39 25 = –3(2x + 1)–63 = –7(8 – p)
p = –1
1. 3.2.
Solving Equations by using Solving Equations by using Distributive Distributive PropertyProperty
3
14
GUIDED PRACTICE
55 + 3x = 8x1. What’s the goal?
– 3x – 3x Get the variables on one side...…and the constants on the other.
…so, if you get rid of 3x3x on the left, you’ll have it.
55 = 5x
Solve.11 = x
or
x = 11
Solving Equations with Solving Equations with Variables on Both Variables on Both Sides*Sides*
*(not taught in Math 7)*(not taught in Math 7)
GUIDED PRACTICE
9x = 12x – 92.
x = 3
–15x + 120 = 15x3.
4 = x
Solving Equations with Solving Equations with Variables on Both Variables on Both SidesSides
Solving Equations with Solving Equations with Variables on Both Variables on Both Sides*Sides*
*(not taught in Math 7)*(not taught in Math 7)
Solving Equations with Solving Equations with Variables on Both Variables on Both Sides*Sides*
*(not taught in Math 7)*(not taught in Math 7)
GUIDED PRACTICE
4. 4a + 5 = a + 11
a = 2
1. Get the variables on one side... …and the constants on the other.
…but, which side for each?
...it doesn’t really matter.
Hint: Move the smaller Hint: Move the smaller variable to the larger variable to the larger variable’s side.variable’s side.
–a –a
3a + 5 = + 11
– 5 – 5
Subtract 55 to isolate the variable.
3a = 6 Solve.
Solving Equations with Solving Equations with Variables on Both Variables on Both SidesSides
Solving Equations with Solving Equations with Variables on Both Variables on Both Sides*Sides*
*(not taught in Math 7)*(not taught in Math 7)
Solving Equations with Solving Equations with Variables on Both Variables on Both Sides*Sides*
*(not taught in Math 7)*(not taught in Math 7)
–6c + 1 = –9c + 7119.
c = 2
120.
n = –8
3n + 7 = 2n –1118.
11 + 3x – 7 = 6x + 5 – 3x
121. 6x + 5 – 2x = 4 + 4x + 1
there are no solutions for x all values of x are solutions
Solving Equations with Solving Equations with Variables on Both Variables on Both SidesSides
Solving Equations with Solving Equations with Variables on Both Variables on Both Sides*Sides*
*(not taught in Math 7)*(not taught in Math 7)
Solving Equations with Solving Equations with Variables on Both Variables on Both Sides*Sides*
*(not taught in Math 7)*(not taught in Math 7)
y = –3w = –18
GUIDED PRACTICE
122. 4(w – 9) = 7w + 18123. 2(y + 4) = –3y – 7
Solving Equations with Solving Equations with Variables on Both Variables on Both SidesSides
Solving Equations with Solving Equations with Variables on Both Variables on Both Sides*Sides*