Download Solving a linear equation

Chapter 3
Solving Linear
Equations
3.1 Solving Equations Using
Addition and Subtraction
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You can solve an equation by using the
transformations below to isolate the
variable on one side of the equation.
When you rewrite an equation using these
transformations, you produce an equation
with the same solutions as the original
equation.
These equations are called equivalent
equations.
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To change, or transform, an equation into
an equivalent equation, think of an
equation as having two sides that are “in
balance.”
Any transformation you apply to an
equation must keep the equation in
balance.
For example, if you if you subtract 3 from
one side of the equation, you must also
subtract 3 from the other side of the
equation.
Transformations that produce
Equivalent Equations
Original
Equation
Add the same x – 3 = 5
# to each
side
Subtract the x + 6 = 10
same # to
each side
Simplify one x = 8 – 3
or both sides
Interchange
the sides
7=x
Equivalent
Equation
Example 1: Solve.
 a) x – 5 = -13
 b) x – 9 = -17
 Example 2: Solve.
 a) -8 = n – (-4)
 b) -11 = n – (-2)
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Linear Equations: The variable is raised to the
first power and does not occur in a denominator,
inside a square root, or inside absolute value
symbols.
Linear Equation
Not a Linear Equation
x+5=9
x2 + 5 = 9
- 4 + n = 2n – 6
|x + 3| = 7
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Example 3: Several record temperature
changes have taken place in Spearfish,
South Dakota. On January 22, 1943, the
temperature in Spearfish fell from 54°F at
9:00am to -4°F at 9:27am. By how many
degrees did the temperature fall?
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Example 4: Match the real-life problem with
an equation.
x – 4 = 16 x + 16 = 4 16 – x = 4
a) You owe $16 to your cousin. You paid x
dollars back and now you owe $4. How much
did you pay back?
b) The temperature was x°F. It rose 16°F and is
now 4°F. What was the original temperature?
c) A telephone pole extends 4 feet below ground
and 16 feet above ground. What is the total
length x of the pole?
3.2 Solving Equations using
Multiplication and Division
Original
Equation
Multiply
each
equation by
the same
nonzero #
Divide each
equation by
the same
nonzero #
x
3
2
4x = 12
Equivalent
Equation
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Example 1: Solve.
a) - 4x = 1
b) 7n = - 35
Example 2: Solve.
a) x  30
5
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b)
f
3
7
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Example 3: Solve.
a)
3
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b)
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 t 9
4
2
10   m
3
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Properties of Equality:
Addition Property:
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Subtraction Property:
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Multiplication Property:
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Division Property:
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3.3 Solving Multi-Step Equations
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Solving a linear equation may require two
or more transformations.
Simplify one or both sides of the equation
(if needed).
Use the inverse operations to isolate the
variable.
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Example 1: Solve.
a) 1
x  6  8
3
b) 1
x  5  10
2
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Example 2: Solve.
a) 7x – 3x – 8 = 24
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b) 2x – 9x + 17 = - 4
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Example 3: Solve.
a) 5x + 3(x + 4) = 28
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b) 4x – 3(x – 2) = 21
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Example 4: Solve.
a) 4x + 12(x – 3) = 28
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b) 2x – 5(x – 9) = 27
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Example 5: Solve.
a) 66   6 ( x  3)
b)
5
3
12  ( x  2)
10
Example 6: A body temperature of 95°F or
lower may indicate the medical condition called
hypothermia. What temperature in the Celsius
scale may indicate hypothermia?
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Example 7: The temperature within
Earth’s crust increases about 30° Celsius
for each kilometer of depth beneath the
surface. If the temperature at Earth’s
surface is 24°C, at what depth would you
expect the temperature to be 114°C?
3.6
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Objective: To find exact and
apporoximate solutions of equations that
contain decimals.
Round-off Error:
Example 1: Solve the equation. Round
to the nearest hundredth.
a) 7.23x + 16.51 = 47.89 – 2.55x
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Example 1: Solve the equation. Round
to the nearest hundredth.
a) 7.23x + 16.51 = 47.89 – 2.55x
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b) 6.6(1.2 – 7.3x) = 16.4x + 5.8
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Example 2: Multiply the equation by a
power of 10 to write an equivalent
equation with integer coefficients. Solve
the equation and round to the nearest
hundredth.
a) 3.11x – 17.64 = 2.02x -5.89
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b) 5.8 + 3.2x = 3.4x – 16.7
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3.7 Formulas and Functions
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Objective: To solve a formula for one of
its variables and rewrite an equation in
function form
Formula: an algebraic expression that
relates two or more real-life quantities.
Example 1: Use the formula for area of
a rectangle – A = lw
a) Find a formula for l in terms of A and w
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b) Use the new formula to find the length
of a rectangle that has an area of 35 sq.
ft. and a width of 7 feet.
Example 2: Solve the temperature formula
C = 5/9(F – 32) for F.
Example 3: a) Solve the simple interest
formula for r.
b) Find the interest rate for an investment
of $1500 that earned $54 in simple
interest in one year
Example 3: Rewrite the equation
3x + y = 4 so that y is a function of x.
Example 4: a) Rewrite the equation
3x + y = 4 so that x is a function of y.
b) Use the result to find x when y = -2, -1,
0 and 1