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Equations and Problem Solving
2.5
Solve each formula in terms of the given variable.
1. 5d – 2g = 9; g
2. 4xy + 3 = 5z; y
3. d(a – b) = c; a
4.
5 1
 (b  c); b
2 2
5. The formula A = 2h(l + w) give the lateral area (A) of a rectangle solid with length (l), width (w), and
height (h). Find h if A=37.4ft2, l- 4.3ft, and w = 6.7 ft.
Like terms – have exactly the same variable factors. The coefficient does
not matter.
Use any of the following symbols to show the like terms below:
  O
3x
-5x2
-7x2y3
3z
4y
x2y
xy2
-2x
Examples
1. 6x – 2 = x +13
2. 4p – 10 = p + 3p – 2p
3. 3(x-4) = 3x – 12
4. 9x + 3x – 10 = 3(3x+x)
SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES
2.4
*Solving equations with variables on both sides is a great way to solve real-life situations such
as cost comparisons.
STEPS
1. Combine like terms on
each side of the equation
(equal sign). Hint: Use the
Distributive Property to
get rid of parentheses.
2. Combine like terms
across the equation/equal
sign. Hint: Use the
Addition or Subtraction
Property of Equality.
Sample A
Sample B
Sample C
6(w – 5) = 3w + 12
18x + 6 = 3(6x + 1)
3(x + 8) + 5x = 2(12 + 4x)
6w – 30 = 3w + 12
18x + 6 = 18x + 3
8x +24 = 24 + 8x
3. Isolate the variable (get
the variable alone). Hint:
Use the Multiplication or
Division Property of
Equality.
4. Simplify
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Distributive Property: For every real number a, b, and c,
a(b + c) = ab + ac
(b + c)a = ba + ca
a(b – c) = ab – ac
(b – c)a = ba – ca
Example:
1.
3(4m – 7)
2.
(3 – 8x)(-1)
COMBINING LIKE TERMS/DISTRIBUTIVE PROPERTY
1.7
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1.
Addition Property Of Equality
Subtraction Property Of Equality
For every real number a, b, and c,
For every real number a, b, and c,
If a = b, then a + c = b + c.
If a = b, then a – c = b – c.
x–3=–8
2. b – 13 = – 24
1. m + 5 = 8
2. 9 + y = 11
1. 72 + 4 – 14 c
2. -13 = 2b – b – 10
3. 5 (y – 3) = 19
4. x/6 – 7 2/3
5. ¼(g – 16) = 7
SOLVING MULTI- STEP EQUATIONS
2.3
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STEPS FOR SOLVING A MULTI-STEP EQUATION
1. Clear the equation of
fractions and decimals.
15 = -3(x – 1) + 9
-2y + 5 + 5y = 14
2. Use the Distributive
Property to remove
parentheses on each side.
Skip (no parentheses)
3. Combine Like Terms
on each side.
4. Undo addition or
subtraction.
5. Undo multiplication
or division.
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1.
3.
Multiplication Property of Equality
Division Property of Equality
For every real number a, b, and c,
For every real number a, b, and c,
If a = b, then ac = bc.
If a = b, then a  b .
m
5
6
2.

d
 20
5
3
x  15
8
SOLVING ONE-STEP EQUATIONS
c
1.
3f = 12
2.
-8 = 5y
3.
-7k = 9
c
2.1
STEPS FOR SOLVING 2-STEP EQUATIONS
1. Use the Addition or
Subtraction Property of
Equality to get the term
with a variable alone on
one side of the equation.
x
 15  12
9
3x – 2 = 4
2. Use the Multiplication
or Division Property of
Equality to write an
equivalent equation in
which the variable has a
coefficient of 1.
1.
3.
10 
m
2
4
x
 5 1
12
SOLVING TWO-STEP EQUATIONS
2.
– a – 5 = -8
4.
8 – 3y = 14
2.2